Dirac Vector Model Research Articles

Interaction Energies and NMR Indirect Nuclear Spin−Spin Coupling Constants in Linear HCN and HNC Complexes

The cooperativity effects on both the electronic energy and NMR indirect nuclear spin-spin coupling constants J of the linear complexes (HCN)n and (HNC)n (n = 1-6) are discussed. The geometries of the complexes were optimized at the MP2 level by using the cc-pVTZ basis sets. The spin-spin coupling constants were calculated at the level of the second-order polarization propagator approximation with use of the local dense basis set scheme based on the cc-pVTZ-J basis sets. We find strong correlations in the patterns of different properties such as interaction energy, hydrogen bond distances, and spin-spin coupling constants for both series of compounds. The intramolecular spin-spin couplings are with two exceptions dominated by the Fermi contact (FC) mechanism, while the FC term is the only nonvanishing contribution for the intermolecular couplings. The latter do not follow the Dirac vector model and are important only between nearest neighbors.

Analysis of long-range NMR spin–spin coupling in polyenes and the π-mechanism

The J-OC-PSP (decomposition of J into Orbital Contributions using Orbital Currents and Partial Spin Polarization) method is applied to analyze NMR spin-spin coupling constants in polyenes, which were calculated using coupled perturbed density functional theory in connection with the B3LYP hybrid functional and a [7s,6p,2d/4s,2p] basis set. The analysis revealed that the pi-mechanism for Fermi contact (FC) spin coupling is based on passive pi-orbital contributions. The pi-orbitals transfer spin information between sigma orbitals (spin-transport mechanism) or increase the spin information of a sigma orbital by an echo effect. The calculated FC(pi) values are rather constant for small polyenes ranging between 3.5-5.5 Hz for a double bond. They decay more slowly with the distance between perturbing and responding nucleus than the sigma contributions to the FC term. The sign of the passive FC(pi) contribution can be assessed from a Dirac vector model. The limits for long-range coupling in a polyene were determined and their practical implications discussed.

Computed Spin−Spin Coupling Constants (1JX-Y) in Molecules HmX−YHnfor X and Y =13C,15N, and31P: Comparisons with Experiment and Insights into the Signs of1JX-Y

One-bond X-Y spin-spin coupling constants ( 1 J X - Y ) for 18 H m X-YH n molecules, with X and Y = 1 3 C, 1 5 N, and 3 1 P, have been computed using the equation-of-motion coupled-cluster singles and doubles method. The molecules investigated include all possible combinations of these three elements bonded with single, double, and triple bonds. The computed coupling constants are in good agreement with experiment over a range that extends from -250 to +200 Hz. With only two exceptions, the sign of the Fermi-contact (FC) term is the same as the sign of 1 J X - Y , but the FC term may or may not be a good quantitative estimate of 1 J X - Y . When reduced spin-spin coupling constants ( 1 K X - Y ) are used for comparing coupling constants involving different atoms, a linear relationship is observed between 1 K X - N and 1 K X - P . The signs of 1 J X - Y for approximately half of the molecules included in this study are exceptions to the Dirac vector model. The recently proposed NMR triplet wave function model has been used to provide insight into the variation of the signs of these one-bond spin-spin coupling constants.

What determines the sign of the Fermi-contact contribution to the NMR spin–spin coupling constant?

The Dirac vector model for the sign of the spin–spin coupling constant ( J) between a pair of atoms states that one-bond coupling constants are positive, two-bond negative, three-bond positive, etc. However, this rule is often violated. In an attempt to understand the sign of J, we propose a new model, the Nuclear magnetic resonance triplet wavefunction model (NMRTWM) which is based on the phases of excited triplet state wavefunctions, and the response of magnetic nuclei to these phases. The model demonstrates that, (1) the excited states which contribute to the coupling constant do so in a regular manner; (2) the sign of an individual contribution is determined by the nodal character of the excited-state wavefunction and the response of the nuclei to the phases of this function; and (3) the sign of the total coupling constant is the result of competing positive and negative contributions from various states. NMRTWM provides a fundamental explanation for both the successes and failures of the Dirac vector model, and can be used to gain insight into some puzzling results for the orientation dependence of F–F coupling between two HF molecules.

Theoretical investigations of the two-bond proton-carbon-13 coupling constants. Angular variations of the couplings involving carboxyl carbon

A formulation has been developed using perturbation theory to evaluate the π-contribution to the nuclear spin coupling constants involving nuclei at least one of which is an unsaturated center. This fromulation accounts for the π-contribution in terms of the core polarization and one-center exchange at the π-center. The formulation developed together with the Dirac vector model and Penney-Dirac bond-order formalisms was employed to calculate the geminal (two-bond) proton coupling constants of carboxyl carbons in α-disubstituted acetic acids. The calculated coupling constants were found to have an orientational dependence. The results of the calculation are in good agreement with the experimental values.

Conformational dependence of vicinal 13C′NCαH coupling constants in peptides: A dirac vector model investigation

AbstractTheoretical calculations of the heteronuclear vicinal coupling constant 3J(13C′NCαH) in peptides have been carried out using the Dirac vector model. The results showed an angular dependence for this coupling constant, which can be expressed in the form 3J(13C′NCαH) = A cos2 θ + B cos θ + C, where A, B, and C are constants and θ is related to the torsional angle ϕ of the peptide backbone. The results of the present calculations are in very good agreement with those obtained using finite perturbation theory at the INDO level of approximation.

Psi‐angle dependence of geminal proton–carbonyl‐13C coupling constants in peptides. A theoretical investigation

AbstractTheoretical calculations of the geminal carbonyl‐13C‐ proton coupling constant, 2J(C′H), in α‐amino acids have been carried out using Dirac Vector model and Penney‐Dirac bond order formulations. The results indicate that the couplings are dependent on the backbone torsion angle psi (ψ) of the amino acid residues in peptides. The meagre available experimental data seem to support the theoretical findings.

Dirac vector model and variational treatments of proton spin-spin coupling across three bonds

The Dirac vector model and variational treatments have been used to study the angular dependence of proton spin couplings across three bonds in fragments of the type H-X-Y-H (X, Y = C, N, O). These couplings can be expressed by means of the relation J(φ) = A + B cos φ + C cos2 φ, where φ is the dihedral angle and A, B and C are constants. It is seen that the behaviour of various fragments towards substituents can be qualitatively predicted by looking at the sign of the constant B.

Spin Polarization in an Aliphatic Fragment: Comparison between Proton Hyperfine Interaction and Spin–Spin Coupling

Expressions are derived for proton hyperfine interaction constants (hfs) in an aliphatic chain arising from a single unpaired pπ electron on the first carbon atom. The method used is the Dirac vector model. Only spin polarization is considered, while electron delocalization is neglected. The resulting expressions are very similar to those derived by Alexander for proton spin–spin coupling. This makes it possible to use experimental spin–spin data for a semiempirical estimate of the contribution of spin polarization to the aliphatic proton hfs. It is found that spin polarization contributes substantially to the hfs of protons removed up to five bonds from the pπ center. The contribution of spin polarization to α-proton hfs is positive and amounts to about 50% of the observed value. For β and γ protons spin polarization contributes, respectively, negative and positive spin densities of the order found experimentally. Deviations from the calculated values are discussed in terms of spin delocalization and conformational effects.

The size and energy of d orbitals in excited states of the sulphur atom

Wave-mechanical calculations are reported for the sulphur atom in varying degrees of ionization. The Dirac vector model is used to set up the self-consistent-field equations appropriate to (1) spectroscopic states involving one or more electrons in d orbitals; (2) valence-states with one or more d electrons; (3) valence-states in which octahedral, square-planar, and distorted tetrahedral hybrids are used. The equations are solved to determine the total atomic energy and the size and shape of the orbitals. Previous less accurate work with use of simple analytical approximations to the orbitals is confirmed and improved. The sizes of the 3s and 3p orbitals change very little, even when the total number of valence electrons changes; but the size of the 3d orbital is very sensitive both to the number of d electrons and the formal charge on the atom. The importance of this is stressed for an understanding of the higher valences of sulphur, and of sulphur, and of other atoms where d electrons may reasonably occur.

Relative signs of six proton coupling constants in isopropenylacetylene. The effect of the acetylenic group on the chemical shift of the protons

The relative signs of the six proton coupling constants in isopropenylacetylene have been determined by multiple resonance techniques. The signs of the long-range couplings agree with those predicted by the Dirac vector model. The geminal ethylenic proton coupling is taken as positive. It is concluded that the methyl group does not interact with the triple bond via a hyperconjugative mechanism. The ethylenic protons are less shielded than in propene, whereas the opposite is expected on the basis of the magnetic anisotropy of the triple bond. This apparent anomaly can be rationalized quantitatively with a model given by Castellano and Lorenc and all four shifts relative to propene are predicted satisfactorily.

Excitation Energy in the Nuclear Spin—Spin Interaction

Valence bond theory, Dirac's vector model, and a perturbation calculation express the coupling constant between nuclear spins as a function of exchange integrals. The approximation of mean excitation energy is not utilized. The singlet states are represented by the product of singlet functions of two electrons in one bond, or singlet functions of four electrons in two bonds, while the triplet states are represented by the product of one triplet function of two electrons in one bond, and of the preceding singlet functions. The different transfer processes of the excitation from a nucleus to the other in the molecule are given.

Derivation of the Dirac Vector Model for a Solid

A cluster expansion derivation of the Dirac vector model, which has important applications in the theory of ferromagnetic and antiferromagnetic materials, is given for the case of a solid with a wave function which is a linear combination of Slater determinants, each corresponding to a different spin configuration. It is assumed that the single-particle wave functions are nonorthogonal, with one particle per lattice site. It is shown that there is no "nonorthogonality catastrophe" and that Carr's condition for the validity of the vector model applies in this case, as he has conjectured. Expressions are given for the direct energy and exchange integral.

Effective Exchange Integral

The magnetic properties of a linear chain of monovalent atoms are investigated from the point of view of perturbation theory. The many-electron wave functions for the system are expanded as linear combinations of determinantal functions which are eigenfunctions of ${S}^{2}$ and ${S}_{z}$. These determinantal functions are constructed from orthonormal one-electron orbitals of the Wannier type so that the nearest neighbor exchange integral is positive definite and approaches zero at large lattice spacings. The secular equation is set up using the method of the Dirac vector model. By means of the Kramers perturbation technique, the interaction of ionic states with those arising from the ground configuration is represented by means of an effective Hamiltonian operator with its associated matrix. The results of this treatment are analogous to those obtained by Paul in that an analytic expression is found for an effective nearest neighbor exchange integral ${J}^{\ensuremath{'}}$. This quantity is represented as the difference between the positive definite exchange integral and additional terms from ionic states. The present treatment defines in a fairly precise manner the type configurations which contribute to this effective exchange integral and the limits for which this parameterization is valid. The results of this analysis are compared with those obtained from recent calculations on a system of six hydrogen atoms.

Spin-Spin Interactions in Nuclear Magnetic Resonance. Contact Contribution

A method for calculating the contact contribution to spin-spin interactions between nuclei in nonaromatic molecules is described. The method is based on the valence bond model. The equivalent Hamiltonian of the Dirac vector model is used for a perturbation calculation in a representation where the total spin of the two electrons in each bond in the molecule is a good quantum number. The Ramsey-Purcell contact term in the interaction is calculated by a double perturbation method where only terms linear in the electron-nuclear interactions are considered but the perturbation is carried to higher order in the exchange integrals. In this way the interaction constants can be obtained explicitly in terms of the exchange integrals, and the calculation of the valence bond wave functions is avoided. It is then possible to see how the signs and magnitudes of the interaction constants depend on these integrals and what the dominant interaction mechanisms are. The perturbation series is evaluated explicitly with the electron nuclear contact interaction and the results are compared with the results of a similar calculation based on Ramsey's closure formula. In this way the objections against the closure procedure can be avoided. It is shown that Ramsey's formula can be used when different average energy separations are used for terms of different order in the expansion. The results of Ramsey for H2 and of Karplus et al. for hydrogen atoms separated by two and three bonds and the justification of the method of calculation used by them are discussed in detail. It is found that only perturbation terms of one type are important in each of the cases discussed by these authors and therefore their procedure can be justified. The results previously obtained by the author for spin-spin interactions in the allyl group are also discussed.

VECTOR MODEL FOR INDIRECT PROTON HYPERFINE INTERACTIONS IN π-ELECTRON RADICALS

Significant information on unpaired electron distributions in molecules, liquids, and solids can sometimes be obtained from magnetic resonance studies of electron-nucleus hyperfine interactions. A particularly important problem in this field involves the proton hyperfine interactions that have been observed in the electron magnetic resonance spectra of a large number of aromatic radicals in solution [1-3]. The problem concerns the theoretical basis for a proposal that the observed hyperfine splitting aN due to aromatic proton N in an aromatic radical can be used to estimate the unpaired spin density ρN at carbon atom N to which proton N is attached, according to the equation [2,4-6] aN = QρN. (1) A semiempirical value of Q derived from observed splittings aN and theoretically calculated spin densities ρN is -30 ± 5 gauss, or -85 ± 15 Mc [2]. Equation (1) has been derived recently using molecular orbital theory with arbitrary π-π configuration interaction [6]. We show here how the simple proportionality in equation (1) may be derived using the Dirac vector model.

On the Configurationsll's

The spectra of the configurations $ll'\mathrm{s}$ are calculated by using Dirac's vector model. A comparison is made with the experimental data of Sc I, Y I, and Cu I.

Diamagnetic Anisotropy and Electronic Structure of Aromatic Molecules

Previous theories of the diamagnetic anisotropy of aromatic molecules, based on the method of molecular orbitals, neglect the correlation effects between electrons. In the present theory the anisotropy is treated from the standpoint of the Heitler-London-Pauling-Slater approximation by the inclusion of ionic terms in the secular equation. By means of an approximate treatment it is shown that correlation effects tend to reduce the diamagnetic susceptibility, the effect being relatively greater in complex molecules than in benzene. The contribution of higher order permutation integrals is also discussed and found to be negligible. The correct approximation lies intermediate between the pure H-L-S-P and pure molecular orbital approximations, but in order to explain the high value of the anisotropy, one is forced to assume that the true situation lies much closer to the molecular orbital viewpoint. The variation of the anisotropy of benzene through all intermediate stages of approximation is discussed by means of a more exact model. All calculations were carried out by treating the permutational degeneracy by means of the Dirac vector model.

Application of the Dirac Vector Model to thed4Configuration

Formulae for the energies of multiplets belonging to ${d}^{4}$ are computed by the use of the vector model as developed by Van Vleck and Serber. Since insufficient data were available for ${d}^{4}$, the configuration $m{d}^{4}\mathrm{ns}$ was used to check the formulae against experiment. Even here the data are rather meager. In order to obtain out of the above configuration the proper energies of the multiplets arising from ${d}^{4}$ alone, it was necessary to subtract from it the exchange energy between the $s$ electron and the ${d}^{4}$ core. The agreement with experiment is, at best, only fair, perhaps because of perturbations from neighboring levels.

Extension of the Dirac Vector Model to Include Several Configurations

Extension of the Dirac Vector Model to Include Several Configurations