Contracted Schrödinger equation in quantum phase‐space

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi‐distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position‐momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.     

PELDOR on an exchange coupled nitroxide copper(II) spin pair

Transition metal ions play an important role in the design of macromolecular architectures as well as for the structure and function of proteins and oligonucleotides, which makes them interesting targets for spectroscopic investigations. In combination with site directed spin labelling, pulsed electron–electron double resonance (PELDOR or DEER) could be a well-suited method for their characterization and localization. Here, we report on the synthesis and full characterization of a copper(II) porphyrin/nitroxide model system bearing an extended π-conjugation between the spin centres and demonstrate the possibility to disentangle the dipolar through space interaction from the through bond exchange coupling contribution even in the presence of orientational selectivity and conformational flexibility. The simulations used are based on the known experimental and spin Hamiltonian parameters and on a structural model as previously employed for similar systems. The mean exchange coupling of +4(1) MHz (antiferromagnetic) is in agreement with the value of |J| = 3(1) MHz determined from room temperature continuous wave electron paramagnetic resonance (EPR). Thus, as long as the pulse excitation bandwidths are large versus the spin–spin coupling, X-band PELDOR measurements in combination with explicit time trace simulations allow for disentangling the sign and magnitude of through bond electron–electron exchange from the through space dipolar interaction D.     

Superconductivity in cuprates: Details of electron phonon coupling

The Bardeen‐Cooper‐Schrieffer (BCS) model explains superconductivity (SC) as due to correlation between electronic momentum and nuclear momentum (phonons) in a free electron gas. The BCS model lacks chemical specificity, however, since the coupling mechanism is left unspecified. After the discovery of high T_C superconductivity in 1986 it was concluded that electron–phonon interactions are insufficient to explain electron pairing. A large part of theoretical research has since been aiming at finding another mechanism that would allow us to consider the superconducting system as a gas of charged free bosons. However, there appears to be no reason to assume free electrons in oxides. In this article the free‐electron criterion is therefore replaced by the criterion that a pair of electrons can move freely between sites without resistance, i.e., without activation energy. Electron pair transfer is treated in a many‐electron real space approach using standard mixed‐valence theories. Mott‐Hubbard‐U is strictly defined, its dependence on breathing mode coordinates analyzed, and the connection between U and the energy gap for superconductivity clarified. d‐wave gap anisotropy is found to be consistent with the general atomic level model presented here. Softening of phonon half‐breathing modes in inelastic neutron scattering (INS) is connected to mixed‐valency. The fundamental vibronic interaction between spin density wave (SDW) and charge density wave (CDW) states leads to a new phase with energy gap and electron pair carriers that can only be the superconducting phase. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Generation of generalized branching diagram spin functions by Schmidt orthogonalization of spin-paired functions

The generalized branching diagram (GBD ) spin representation is defined as the method of sequentially coupling together a number of subsystem spin eigenfunctions using the general rules of angular momentum coupling. It is shown that any GBD representation may also be obtained by Schmidt orthogonalizing a set of cannonical spin–paired (SP ) functions, provided the SP basis is suitably ordered. The ordering procedure used is well suited to computer implementation. This is a generalization of results known in the literature for the Yamanouchi–Kotani and for the Serber spin representations.     

Atomic shells from the electron localizability in momentum space

The electron localizability indicator in momentum space is proposed as a functional of the same‐spin momentum pair density. This functional yields a discrete distribution of values, which are proportional to the charge needed to form a fixed very small fraction of a same‐spin electron pair. It resolves all atomic shells for the examined atoms (Li–Kr) with reasonable occupation numbers, especially in the valence region. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Electron spin exchange processes in strongly coupled spin triads

The complexes of Cu2+ hexafluoroacetylacetonate with two pyrazol-substituted nitronyl nitroxides are the choice systems to study the spin dynamics of strongly exchange-coupled spin triads. The large values of exchange coupling (ca. 100 cm-1) and high-resolution electron paramagnetic resonance (EPR) at Q- and W-bands (35 and 94 GHz) allowed us to observe and interpret specific characteristics of these systems. An electron spin exchange process has been found between different multiplets of the spin triad, which manifests itself as a significant shift of the EPR line position with temperature. We propose that the spin exchange process is caused by the modulation of exchange interaction between copper and nitroxides by lattice vibrations. The estimations of the rate of exchange process and model calculations essentially support the observed phenomena. The studied characteristics of strongly coupled spin triads explain previously obtained results, agree with literature, and should be accounted for in future investigations of similar spin systems.     

Quantum Chemical Spin Densities for Radical Cations of Photosynthetic Pigment Models

The spin densities of radical cations of magnesium porphyrin, magnesium chlorine and a truncated chlorophyll a model are calculated with density‐functional theory and multiconfigurational quantum chemical methods. The latter serve as a reference for approximate density‐functional theory which yields spin densities that may suffer from the self‐interaction error. We carried out complete active space self‐consistent field calculations with increasing active orbital spaces to systematically converge qualitatively correct spin densities. In particular, for the magnesium chlorine and chlorophyll a model radical cations, this is not easy to achieve because of the lower symmetry compared to magnesium porphyrin. Strategies had to be employed which allowed us to consider very large active orbital spaces. We explored restricted active space self‐consistent field and density‐matrix renormalization group calculations. Based on these reference data, we assessed the accuracy of different density‐functional approximations. We show that in particular, exchange–correlation model potentials with correct asymptotic behavior yield good spin densities, and we find, in agreement with previous studies on different classes of compounds, that hybrid functionals systematically increase spin‐polarization effects with increasing amounts of exact exchange. Our results provide a starting point for investigations of spin densities of more complex systems such as the hinge model for the primary electron donor in photosystem II.     

Trends in exchange coupling for trimethylenemethane-type Bis(semiquinone) biradicals and correlation of magnetic exchange with mixed valency for cross-conjugated systems

A magnetostructural correlation (conformational electron spin exchange modulation) within an isostructural series of biradical complexes is presented. X-ray crystal structures, variable-temperature electron paramagnetic resonance spectroscopy, zero-field splitting parameters, and variable-temperature magnetic susceptibility measurements were used to evaluate molecular conformation and electron spin exchange coupling in this series of molecules. Our combined results indicate that the ferromagnetic portion of the exchange couplings occurs via the cross-conjugated pi-systems, while the antiferromagnetic portion occurs through space and is equivalent to incipient bond formation. Thus, molecular conformation controls the relative amounts of ferro- and antiferromagnetic contributions to exchange coupling. In fact, the exchange parameter correlates with average semiquinone ring torsion angles via a Karplus-Conroy-type relation. Because of the natural connection between electron spin exchange coupling and electronic coupling related to electron transfer, we also correlate the exchange parameters in the biradical complexes to mixed valency in the corresponding quinone-semiquinone radical anions. Our results suggest that delocalization in the cross-conjugated, mixed-valent radical anions is proportional to the ferromagnetic contribution to the exchange coupling in the biradical oxidation states.     

Electron localizability indicators ELI–D and ELIA for highly correlated wavefunctions of homonuclear dimers. I. Li2, Be2, B2, and C2

Electron localizability indicators based on the parallel‐spin electron pair density (ELI–D) and the antiparallel‐spin electron pair density (ELIA) are studied for the correlated ground‐state wavefunctions of Li₂, Be₂, B₂, and C₂ diatomic molecules. Different basis sets and reference spaces are used for the multireference configuration interaction method following the complete active space calculations to investigate the local effect of electron correlation on the extent of electron localizability in position space determined by the two functionals. The results are complemented by calculations of effective bond order, vibrational frequency, and Laplacian of the electron density at the bond midpoint. It turns out that for Li₂, B₂, and C₂ the reliable topology of ELI–D is obtained only at the correlated level of theory. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

The Stern-Gerlach experiment and the effects of spin relaxation

The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.     

Photoinduced Electron Transfer within a Zinc Porphyrin–Cyclobis(paraquat‐p‐phenylene) Donor–Acceptor Dyad

Understanding the mechanism of efficient photoinduced electron‐transfer processes is essential for developing molecular systems for artificial photosynthesis. Towards this goal, we describe the synthesis of a donor–acceptor dyad comprising a zinc porphyrin donor and a tetracationic cyclobis(paraquat‐p‐phenylene) (CBPQT⁴⁺) acceptor. The X‐ray crystal structure of the dyad reveals the formation of a dimeric motif through the intermolecular coordination between the triazole nitrogen and the central Zn metal of two adjacent units of the dyad. Photoinduced electron transfer within the dyad in MeCN was investigated by femtosecond and nanosecond transient absorption spectroscopy, as well as by transient EPR spectroscopy. Photoexcitation of the dyad produced a weakly coupled ZnP^+.–CBPQT^3+. spin‐correlated radical‐ion pair having a τ=146 ns lifetime and a spin–spin exchange interaction of only 0.23 mT. The long radical‐ion‐pair lifetime results from weak donor–acceptor electronic coupling as a consequence of having nine bonds between the donor and the acceptor, and the reduction in reorganization energy for electron transfer caused by charge dispersal over both paraquat units within CBPQT^3+..     

High‐spin behavior of molecular crystals and extended π systems

The qualitative rules for the existence of high‐spin ground states in extended systems and molecular crystals are examined here on a firmer theoretical footing. Extended systems have been categorized into three groups, namely, type I, type II, and type III, depending on the type of bonding interactions. The general form of the spin Hamiltonian operators have been written down. The active spaces have been restricted to the minimum size for each of these three types of spin systems. The zeroth‐order state vectors and the Hartree–Fock ground‐state energies have been identified for unit species of each type. The extended system Hamiltonian operators are further truncated in such a way that only the nearest‐neighbor interactions are retained. Expressions have been derived for the energy gap from a molecular orbital approach. The relatively small effects of electron correlation on the energy gaps have been estimated for the type I systems, which belong to the systems of solid‐state physics. In particular, it has been shown that for the type I systems the singlet–triplet gap, and hence the ferromagnetic coupling constant, primarily depends upon the difference of one‐electron kinetic energies and not on the two‐electron exchange integrals. This result agrees with the concept of kinetic exchange that was introduced in the context of a resonating valence‐bond formalism. Type II systems are exemplified by extended systems that can be prepared from conjugated molecules while organic molecular crystals form examples of type III species. For these systems, however, the Coulomb exchange interaction has been shown to dominate the energy gap. A quick review of the Heisenberg spin Hamiltonian for the H₂ molecule is sufficient to point out that the sign of the calculated ferromagnetic coupling constant depends on the method of calculation, the nature of the basis set, and the bond length. This is amply supported by ab initio calculations on this species. Numerical data have also been obtained from computations on m‐phenylene‐coupled nitroxy radicals and stacks of α‐nitronyl nitroxide, but these calculations have been based on a semiempirical quantum chemical methodology (INDO) since some of the species involved are exceedingly large. Computed energy gaps are in good agreement with experimental and other theoretical (AM1, PM3) results. Nevertheless, for the dimer, trimer, tetramer, and pentamer of the type II specimen, the important π orbitals are far from being degenerate. The quantitative results clearly deviate from the criterion of degeneracy that was suggested from qualitative theories for the existence of a high‐spin ground state. Therefore, the criteria for the existence of high spins have been reformulated in terms of the monomer orbitals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 308–324, 2000     

Does the Sign of the Cu–Gd Magnetic Interaction Depend on the Number of Atoms in the Bridge?

Several theoretical investigations with CASSCF methods confirm that the magnetic behavior of Cu–Gd complexes can only be reproduced if the 5d Gd orbitals are included in the active space. These orbitals, expected to be unoccupied, do present a low spin density, which is mainly due to a spin polarization effect. This theory is strengthened by the experimental results reported herein. We demonstrate that Cu–Gd complexes characterized by Cu–Gd interactions through single‐oxygen and three‐atom bridges consisting of oxygen, carbon, and nitrogen atoms, present weak ferromagnetic exchange interactions, whereas complexes with bridges made of two atoms, such as the nitrogen–oxygen oximato bridge, are subject to weak antiferromagnetic exchange interactions. Therefore, a bridge with an odd number of atoms induces a weak ferromagnetic exchange interaction, whereas a bridge with an even number of atoms supports a weak antiferromagnetic exchange interaction, as observed in pure organic compounds and also, as in this case, in metal–organic compounds with an active spin polarization effect.     

Reactivity Controlling Factors for an Aromatic Carbon‐Centered σ,σ,σ‐Triradical: The 4,5,8‐Tridehydroisoquinolinium Ion

The chemical properties of the 4,5,8‐tridehydroisoquinolinium ion (doublet ground state) and related mono‐ and biradicals were examined in the gas phase in a dual‐cell Fourier‐transform ion cyclotron resonance (FT‐ICR) mass spectrometer. The triradical abstracted three hydrogen atoms in a consecutive manner from tetrahydrofuran (THF) and cyclohexane molecules; this demonstrates the presence of three reactive radical sites in this molecule. The high (calculated) electron affinity (EA=6.06 eV) at the radical sites makes the triradical more reactive than two related monoradicals, the 5‐ and 8‐dehydroisoquinolinium ions (EA=4.87 and 5.06 eV, respectively), the reactivity of which is controlled predominantly by polar effects. Calculated triradical stabilization energies predict that the most reactive radical site in the triradical is not position C4, as expected based on the high EA of this radical site, but instead position C5. The latter radical site actually destabilizes the 4,8‐biradical moiety, which is singlet coupled. Indeed, experimental reactivity studies show that the radical site at C5 reacts first. This explains why the triradical is not more reactive than the 4‐dehydroisoquinolinium ion because the C5 site is the intrinsically least reactive of the three radical sites due to its low EA. Although both EA and spin–spin coupling play major roles in controlling the overall reactivity of the triradical, spin–spin coupling determines the relative reactivity of the three radical sites.     

Electron pair density information measures in atomic systems

Shannon entropies of the pair density, conditional entropies, and mutual information are studied in position and in momentum space for ground state neutral atoms and selected excited states at the Hartree‐Fock level. We show that the mutual information, a measure of correlation, is larger in position space than in momentum space. This result also holds for a mutual information defined in terms of the exchange density; however, these quantities display much more structure than the corresponding ones based on the pair densities. The interpretation of this behavior is that exchange effects are smaller in momentum space than in position space in these systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Maximum Spin Polarization in Chromium Dimer Cations as Demonstrated by X‐ray Magnetic Circular Dichroism Spectroscopy

X‐ray magnetic circular dichroism spectroscopy has been used to characterize the electronic structure and magnetic moment of Cr₂⁺. Our results indicate that the removal of a single electron from the 4sσ_g bonding orbital of Cr₂ drastically changes the preferred coupling of the 3d electronic spins. While the neutral molecule has a zero‐spin ground state with a very short bond length, the molecular cation exhibits a ferromagnetically coupled ground state with the highest possible spin of S=11/2, and almost twice the bond length of the neutral molecule. This spin configuration can be interpreted as a result of indirect exchange coupling between the 3d electrons of the two atoms that is mediated by the single 4s electron through a strong intraatomic 3d‐4s exchange interaction. Our finding allows an estimate of the relative energies of two states that are often discussed as ground‐state candidates, the ferromagnetically coupled ¹²Σ and the low‐spin ²Σ state.     

All‐electron relativistic computations on the low‐lying electronic states,bond length,and vibrational frequency of CeF diatomic molecule with spin–orbit coupling effects

Ab initio all‐electron computations have been carried out for Ce⁺ and CeF, including the electron correlation, scalar relativistic, and spin–orbit coupling effects in a quantitative manner. First, the n‐electron valence state second‐order multireference perturbation theory (NEVPT2) and spin–orbit configuration interaction (SOCI) based on the state‐averaged restricted active space multiconfigurational self‐consistent field (SA‐RASSCF) and state‐averaged complete active space multiconfigurational self‐consistent field (SA‐CASSCF) wavefunctions have been applied to evaluations of the low‐lying energy levels of Ce⁺ with [Xe]4f¹5d¹6s¹ and [Xe]4f¹5d² configurations, to test the accuracy of several all‐electron relativistic basis sets. It is shown that the mixing of quartet and doublet states is essential to reproduce the excitation energies. Then, SA‐RASSCF(CASSCF)/NEVPT2 + SOCI computations with the Sapporo(‐DKH3)‐2012‐QZP basis set were carried out to determine the energy levels of the low‐lying electronic states of CeF. The calculated excitation energies, bond length, and vibrational frequency are shown to be in good agreement with the available experimental data. © 2018 Wiley Periodicals, Inc.