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.     

Matrix elements between spin-bonded functions in a hole-particle formalism

Closed formulas are given for Hamiltonian matrix elements between spin-bonded functions in a holeparticle formalism. The derivation is based on Wick's theorem and the use of Jucys diagrams. The final formulas are only slightly more complicated than those for bonded functions in a particle formalism.     

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

The explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin-adapted N-electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many-body time-independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line-up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.     

Matrix element factorization in the unitary group approach for configuration interaction calculations

Techniques of diagrammatic spin algebra are employed to derive segment factorization formulas for spin-adapted matrix elements of one- and two-electron excitation operators. The spin-adapted basis is formed by the Yamanouchi–;Kotani geneological coupling method, and therefore constitutes an irreducible basis of the unitary group U(N), as prescribed by Gel'fand and Tsetlin. Several features distinguish this paper from similar work that has recently been published. First, intermediate steps in the derivation of each segment factor are fully documented. Comprehensive tables list the spin diagrams and phases that contribute to the possible segment factors. Second, a special effort has been made to distinguish between those parts of a segment factor that can be ascribed to a spin diagram and those parts which arise from the orbitals. The results of this paper should thus be useful for those who wish to extend diagrammatic spin algebra to evaluation of matrix elements for states built from nonorthogonal orbitals. Third, a novel graphical method has been introduced to keep track of phase changes that are induced by line up permutations of creation and annihilation operators. This technique may be useful for extension of our analysis to higher excitations. The necessary concepts of second quantization and diagrammatic spin algebra are developed in situ, so the present derivation should be accessible to those who have little prior knowledge of such methods.     

Geometric approximation to nuclear spin–spin coupling constants in the water molecule

The geometric aproximation is used within the framework of triple perturbation theory to evaluate the contributions to nuclear spin–spin coupling constants in the water molecule provided by the Fermi contact, the spin–orbit, and the spin–dipolar interactions. The results, obtained with SCF wave functions expanded over Gaussian basis sets of increasing quality, are compared with corresponding coupled Hartree–Fock estimates. The limits of the geometric approximation to coupling constants are discussed.     

Characterization of bond and current correlation structures in a molecular many-electron wave function

A method has been developed to analyzed the bond and current correlation structures of a molecular many-electron wave function. It is shown that the second order density matrix contains information about the bond and current correlations in its off-diagonal components with respect to the indices of orbital basis functions. We break down the off-diagonal correlation functions into five kinds: charge, spin scalar, spin quadrupole, charge spin, and spin polar correlation functions. For a real wave function, the four correlation functions, except for the spin polar one, have only symmetric–symmetric and antisymmetric–antisymmetric components. The former components give site–bond and bond–bond correlations of charges and spins, while the latter components give current–current correlations of charges and spins. The spin polar correlation function has only symmetric–antisymmetric components that give site–current and bond–current correlations of spins. The five off-diagonal correlation functions are expressed in terms of the off-diagonal components of the second order density matrix. The linked off-diagonal correlation functions are defined in that they give dynamical bond and current correlations. The method is applied to the analyses of the bond and current correlations in the low lying exact eigenstates of the PPP Hamiltonian of benzene.     

Symmetry components analysis of nuclear spin–spin coupling constants

Molecular symmetry properties are used to define “normal” spin–spin coupling constants corresponding to some irreducible representations of the symmetry point group of the molecule. The relationship between these normal coupling constants and the measured ones is established in closed form for the most common cases. The Ramsey perturbation formula is analysed into symmetry components by means of the Winger–Eckart theorem. Both contributions predicted by the molecular-orbital method, i. e. direct coupling via σ electrons and indirect coupling via σ–π interaction are studied. Numerical calculations for the coupling constants of ethane, ethylene and acetylene were carried out without the mean excitation energy approximation by using SCF ? MO wave functions; overlap between atomic orbitals is systematically taken into account by calculating coupling constants. Theoretical and experimental results are compared in terms of symmetry components.     

Synthesis of 15‐, 16‐, and 17‐Membered Bis‐C‐Pivot Macrocycles Consisting of N2O2 Donor‐Type Crown Ethers and Dialkyl Phosphonates

Bis‐C‐pivot macrocycles containing aminophosphonate functions ( 5–10 ) have been synthesized and characterized by elemental analysis, FTIR, MS, 1D ¹H, ¹³C and ³¹P NMR, and 2D HETCOR techniques. The phosphorylation reaction of dibenzo‐bis‐imino crown ethers ( 1–4 ) with dimethyl and diethyl phosphite used here has the potential to provide bis‐C‐pivot macrocycles ( 5–10 ), which possess two stereogenic C‐centers giving rise to diastereoisomers (meso and racemic). Detailed spectral assignments for the meso and racemic forms of the compounds are reported on the basis of chemical shifts, signal intensities, spin–spin coupling constants, and splitting patterns. The bis‐C‐pivot macrocycles ( 5–10 ) may serve as a potential new class of supramolecular host molecules.     

Nonrigid water octamer: Computations with the 8-cube

A 8-cube model of the fully nonrigid water octamer is considered within the 8-dimensional hyperoctahedral wreath product group with 10,321,920 operations and 185 irreducible representations by employing computational and mathematical techniques. For the two lowest-lying isomers of (H₂O)₈ with D_2d and S₄ symmetries of a rigid (H₂O)₈, correlation tables and nuclear spin statistics are constructed for the tunneling splittings of the rotational levels are computed by a computational matrix polynomial generating function technique combined with Möbius inversion, and the relationship to the 8-cube multinomials are pointed out. Multinomial generating functions combined with the induced representation techniques are employed to compute and construct the nuclear spin species, nuclear spin statistical weights and tunneling splittings of rovibronic levels. We have also computed the spin statistical weights and tunneling splittings of the rotational levels for a semirigid water octamer within the wreath product O_h[S₂] consisting of 12,288 operations.     

Unrestricted-Hartree–Fock wave functions approximating low-lying covalent states of ring π systems

We show that spin projected unrestricted-Hartree–Fock (PUHF ) wave functions are able to well approximate some low-lying covalent states of ring π systems. The UHF wave functions belong to either the axial or torsional spin density wave class. Their spin structures are found to be approximations for the spin correlation structures of the corresponding exact wave functions. The PUHF wave functions become close to exact eigenstates with homopolar valence bond characters in the strong correlation limit.     

Macroscopic methods: Magnetic,optical, and calorimetric techniques

The recent effervescent impetus in the spin transition research field is associated with the prodigious multidisciplinary effort that is carried out by different research groups all over the world. Indeed, more and more groups are working in this exciting research field. Nevertheless, this renewed interest also reflects the evolution of the characterization techniques that have been used since the earliest studies of these molecular-based switchable materials. Indeed, we have passed from traditional characterization methods such as UV–vis and Mössbauer spectroscopy or torque magnetometers to the most advanced techniques that are even capable to monitor a spin transition in a single molecule (e.g., X-ray diffraction, X-ray photoelectron spectroscopy, scanning tunneling microscopy, etc.). In this study, we will focus on the basis and the evolution of three critical macroscopic tools, i.e. the optical, magnetic, and calorimetric characterization methods.     

Nuclear reactivity indices in the context of spin polarized density functional theory

In this work, the nuclear reactivity indices of density functional theory have been generalized to the spin polarized case and their relationship to electron spin polarized indices has been established. In particular, the spin polarized version of the nuclear Fukui function has been proposed and a finite difference approximation has been used to evaluate it. Applications to a series of triatomic molecules demonstrate the ability of the new functions to predict the geometrical changes due to a change in the spin multiplicity. The main equations in the different ensembles have also been presented.     

Ab initio Hartree–Fock and multiple-scattering Xα calculation of the g factors for CuF2

Ab initio Hartree–Fock and multiple-scattering wave functions are calculated for linear CuF₂. These wave functions are used to calculate the spin–orbit coupling in a new way where the neglect of two- and many-center terms is avoided and where experimental or calculated spin–orbit coupling constants for the atomic ions are used. The calculated value of g is too small by the MS Xα method and too large by the ab initio method, indicating too much 3d–L interaction in the MS Xα case and too little in the ab initio case.     

How deeply are core electrons perturbed when valence electrons are spin polarized? The case study of transition metal compounds

The ferromagnetic and antiferromagnetic wave functions of the KMnF₃ perovskite have been evaluated quantum-mechanically by using an all electron approach and, for comparison, pseudopotentials on the transition metal and the fluorine ions. It is shown that the different number of α and β electrons in the d shell of Mn perturbs the inner shells, with shifts between the α and β eigenvalues that can be as large as 6 eV for the 3s level, and is far from negligible also for the 2s and 2p states. The valence electrons of F are polarized by the majority spin electrons of Mn, and in turn, spin polarize their 1s electrons. When a pseudopotential is used, such a spin polarization of the core functions of Mn and F can obviously not take place. The importance of such a spin polarization can be appreciated by comparing (i) the spin density at the Mn and F nuclear position, and then the Fermi contact constant, a crucial quantity for the hyperfine coupling, and (ii) the ferromagnetic–antiferromagnetic energy difference, when obtained with an all electron or a pseudopotential scheme, and exploring how the latter varies with pressure. This difference is as large as 50% of the all electron datum, and is mainly due to the rigid treatment of the F ion core. The effect of five different functionals on the core spin polarization is documented.     

Spin–orbit coupling in the intersystem crossing of the ring-opened oxirane biradical

The intersystem crossing (ISC ) between the lowest triplet and singlet states occurring in the reaction of atomic oxygen with ethylene was studied. The importance of spin–orbit coupling (SOC ) in oxirane biradicals (?R′R″—CRR*—?) is stressed through calculations where the spin–orbit matrix elements over the full Breit–Pauli SOC operator has been obtained in the singlet–triplet crossing region. The calculations are performed with a multiconfigurational linear response approach, in which the spin–orbit couplings are obtained from triplet response functions using differently correlated singlet-reference-state wave functions. Computational results confirm earlier semiempirical predictions of the spin–orbit coupling as an important mechanism behind the ring opening of oxiranes and addition of oxygen O(³P) atoms to alkenes. © 1995 John Wiley & Sons, Inc.     

Inclusion of hydrogen p orbitals in the semiempirical calculation of NMR parameters. I. Influence on the FPT INDO spin–spin coupling constants

Hydrogen 2p orbitals have been introduced into the basis set to calculate the Fermi contact term of spin–spin coupling constants using the FPT INDO method. Different coupling constants show different sensitivity to these hydrogen polarization functions. Some improvements are found for molecules containing N or F. Calculations of proton-proton geminal coupling constants give more negative results than those of FPT INDO , yielding a better agreement with experimental values. The π-transmission mechanism is notably exaggerated.     

Non‐empirical calculations of NMR indirect carbon–carbon coupling constants: 2. Strained polycarbocycles

High‐level non‐empirical calculations of carbon–carbon spin–spin coupling constants in a series of strained polycarbocycles have been carried out, in excellent agreement with available experimental data. The utmost importance of electronic correlation effects in this case has been demonstrated and it has been shown that the Second‐Order Polarization Propagator Approach (SOPPA) is an adequate method to account for those effects. It has been demonstrated that the most reliable basis sets to calculate J(C,C) at the SOPPA level are the correlation‐consistent basis sets of Dunning and co‐workers augmented with inner core s‐functions or decontracted in their s‐parts. The nature of the unusual bridgehead–bridgehead bonds in bicyclobutane and propellane in terms of s‐characters of bonding hybrids and also the hybridization effects in spiropentane are discussed based on the arguments derived from the current calculations of J(C,C) in the title compounds. The values of the unknown J(C,C) in propellane and spiropentane are predicted with high reliability. Copyright © 2003 John Wiley & Sons, Ltd.     

Inclusion of hydrogen p orbitals in the semiempirical calculation of NMR parameters. III: INDO CHF calculations of orbital and dipolar contributions to spin–spin coupling constants involving protons

The self-consistent perturbation theory is used to calculate noncontract contributions to spin–spin coupling constants involving protons. Molecular wave functions were obtained with a modified version of the INDO method which includes hydrogen 2p orbitals in its basis set. It is found that in many cases the orbital and dipolar terms are by no means negligible, being particularly important in geminal H? H couplings. Results reported in this paper for this type of coupling, reproduce experimental trends in the series CH₄, NH₃, and OH₂. In general, noncontact terms are found to decrease as the number of bonds separating the interacting nuclei increases.     

Relativistic corrections in the LCGTO–local-spin-density method. I. Atomic bases for transition metals

Compact orbital GTO basis sets optimized with a nonrelativistic Hamiltonian, then decontracted, are utilized in an SCF treatment with a quasi-relativistic scalar Hamiltonian including the mass–velocity and one-electron Darwin operators. Ionization and excitation energies, orbital energies, and radial mean values obtained from different expansion patterns have been tested in atomic calculations for Ag and generalized for Cu and Au atoms. The one-electron spin–orbit operator is also used in an SCF treatment. Spin–orbit coupling energies are calculated for Cu, Ag, and Au atoms.