On the quantum origin of few response properties.

In modern physics, the entanglement between quantum states is a well-established phenomenon. Going one step forward, one can conjecture the likely existence of an entanglement between excitations of one-particle quantum states. Working with a density matrix that is well defined within the polarization propagator formalism, together with information theory, we found that the quantum origin of, at least, few molecular response properties can be described by the entanglement between two pairs of virtual excitations of molecular orbitals (MOs). With our model, we are able to bring new insights into the electronic mechanisms that are behind the transmission, and communication, of the effects of a given perturbation to the whole electronic system described by the Hamiltonian of an unperturbed quantum system. With our entanglement model, we analyzed the electronic origin of the Karplus rule of nuclear magnetic resonance spectroscopy, a well-known empirical phenomenon, and found that this rule is straightforwardly related to the behavior of entangled MO excitations. The model compound used to show it is the H2O2 molecule.

Polarization propagator theory and the entanglement between MO excitations.

Entanglement is at the core of quantum physics and so, one may conjecture that it should have some influence on atomic and molecular response properties. The usual way of treating entanglement is by applying information theory via the von Newman entropy. Given that the principal propagator is the operator that contains the physical information that arises due to the transmission of the effects of two external perturbations through the electronic framework of a quantum system, it should have in it the information necessary to quantify the likely entanglement among molecular orbital excitations. In this article we first propose a proper density matrix and from it, the way to quantify entangled excitations by using information theory. The NMR J-couplings are among the best candidates to learn about the potentialities of this formalism. We applied this new tool to analyze the famous Karplus rule and found a relationship between the dihedral angular dependence and the entanglement. We also found that the entangled excitations are related to electron correlation. The new formalism can be applied to all other response properties.

IPPP-CLOPPA Analysis of the Influence of the Methylation on the Potential Energy and the Molecular Polarizability of the Hydrogen Bonds in the Cytosine-Guanine Base Pair.

The IPPP-CLOPPA method is applied to investigate the influence of a methyl group on the energy of the hydrogen bonds and the potential energy curve of the bridge protons in model compounds, which mimic the methylated and unmethylated cytosine-guanine base pairs. On the same grounds, this influence on the polarizability of the intermolecular hydrogen bonds of these compounds is also addressed, in order to determine whether this linear response property provides a significant proof of the electronic mechanisms that affect the stabilization of the hydrogen bonds. Results obtained show that the methyl electronic system delocalizes on the hydrogen bond region, and changes of these intermolecular hydrogen bonds are due to this effect of delocalization.

Role of Spin-Dependent Terms in the Relationship among Nuclear Spin-Rotation and NMR Magnetic Shielding Tensors.

The broadly accepted procedure to obtain the experimental absolute scale of NMR magnetic shieldings, σ, is well-known for nonheavy atom-containing molecules. It was uncovered more than 40 years ago by the works of Ramsey and Flygare. They found a quite accurate relationship among σ and the nuclear spin-rotation constants. Its relativistic extension was very recently proposed, although it has an intrinsic weakness because a new SO-S two-component term needs to be considered. We show how to overcome this problem. We found that (νYS - νYatom,S) generalizes the SO-S term, where νYS = ⟨⟨[((r - rY) × α)/(|r - rY|3)]; S(4)⟩⟩, r - rY is the electron position with respect to the position of nucleus Y, and S(4) is the four-component total electron spin. When including this new term, one finds that the best of our relativistic Flygare-like models fits quite well with the results of the most accurate method available at the moment. We also show that the difference among the parallel component of σ(Xe) in XeF2 and σ(Xe) of the free atom is almost completely described by that new term.

Toward an absolute NMR shielding scale using the spin-rotation tensor within a relativistic framework.

One of the most influential articles showing the best way to get the absolute values of NMR magnetic shieldings, σ (non-measurables) from both accurate measurements and theoretical calculations, was published a long time ago by Flygare. His model was shown to break down when heavy atoms are involved. This fact motivated the development of new theories of nuclear spin-rotation (SR) tensors, which consider electronic relativistic effects. One was published recently by some of us. In this article we take another step further and propose three different models that generalize Flygare's model. All of them are written using four-component relativistic expressions, though the two-component relativistic SO-S term also appears in one. The first clues for these developments were built from the relationship among σ and the SR tensors within the two-component relativistic LRESC model. Besides, we had to introduce a few other well defined assumptions: (i) relativistic corrections must be included in a way to best reproduce the relationship among the (e-e) term (called "paramagnetic" within the non-relativistic domain) of σ and its equivalent part of the SR tensor, (ii) as happens in Flygare's rule, the shielding of free atoms shall be included to improve accuracy. In the highest accurate model, a new term known as Spin-orbit due to spin, SO-S (in this mechanism the spin-Zeeman Hamiltonian replaces the orbital-Zeeman Hamiltonian), is included. We show the results of the application of those models to halogen containing linear molecules.

Theoretical study of the relativistic molecular rotational g-tensor.

An original formulation of the relativistic molecular rotational g-tensor valid for heavy atom containing compounds is presented. In such formulation, the relevant terms of a molecular Hamiltonian for non-relativistic nuclei and relativistic electrons in the laboratory system are considered. Terms linear and bilinear in the nuclear rotation angular momentum and an external uniform magnetic field are considered within first and second order (relativistic) perturbation theory to obtain the rotational g-tensor. Relativistic effects are further analyzed by carrying out the linear response within the elimination of the small component expansion. Quantitative results for model systems HX (X=F, Cl, Br, I), XF (X=Cl, Br, I), and YH(+) (Y=Ne, Ar, Kr, Xe, Rn) are obtained both at the RPA and density functional theory levels of approximation. Relativistic effects are shown to be small for this molecular property. The relation between the rotational g-tensor and susceptibility tensor which is valid in the non-relativistic theory does not hold within the relativistic framework, and differences between both molecular parameters are analyzed for the model systems under study. It is found that the non-relativistic relation remains valid within 2% even for the heavy HI, IF, and XeH(+) systems. Only for the sixth-row Rn atom a significant deviation of this relation is found.

Breit interaction effects in relativistic theory of the nuclear spin-rotation tensor.

In this work, relativistic effects on the nuclear spin-rotation (SR) tensor originated in the electron-nucleus and electron-electron Breit interactions are analysed. To this end, four-component numerical calculations were carried out in model systems HX (X=H,F,Cl,Br,I). The electron-nucleus Breit interaction couples the electrons and nuclei dynamics giving rise to a purely relativistic contribution to the SR tensor. Its leading order in 1/c is of the same value as that of relativistic corrections on the usual second order expression of the SR tensor considered in previous work [I. A. Aucar, S. S. Gómez, J. I. Melo, C. G. Giribet, and M. C. Ruiz de Azúa, J. Chem. Phys. 138, 134107 (2013)], and therefore it is absolutely necessary to establish its relative importance. For the sake of completeness, the corresponding effect originating in the electron-electron Breit interaction is also considered. It is verified that in all cases these Breit interactions yield only very small corrections to the SR tensors of both the X and H nuclei in the present series of compounds. Results of the present work strongly suggest that in order to achieve experimental accuracy in the theoretical study of the SR tensor both electron-nucleus and electron-electron Breit effects can be safely neglected.

CLOPPA analysis of the molecular polarizability and the energy of strong intramolecular hydrogen bonds: resonance assisted?

The contributions from localized orbitals within the polarization propagator approach (CLOPPA) method is applied to investigate the influence of the π system on the polarizability of the intramolecular hydrogen bond of malonaldehyde, in order to address if this linear response property provides evidence for the existence of the resonance assisted HB (RAHB) mechanism. On the same grounds, this influence on the energy of the hydrogen bond and the potential energy curve of the proton are also evaluated. Results obtained seem to confirm that the π system delocalizes on the hydrogen bond region, and this effect of delocalization is partially responsible for the unusual strength of this intramolecular hydrogen bond.

Theoretical study of the nuclear spin-molecular rotation coupling for relativistic electrons and non-relativistic nuclei.

A theoretical study of the relation between the relativistic formulation of the nuclear magnetic shielding and spin-rotation tensors is presented. To this end a theoretical expression of the relativistic spin-rotation tensor is formulated, considering a molecular Hamiltonian of relativistic electrons and non-relativistic nuclei. Molecular rotation effects are introduced considering the terms of the Born-Oppenheimer decomposition, which couple the electrons and nuclei dynamics. The loss of the simple relation linking both spectral parameters in the non-relativistic formulation is further analyzed carrying out a perturbative expansion of relativistic effects by means of the linear response within the elimination of the small component approach. It is concluded that relativistic effects on the spin-rotation tensor are less important than those of the nuclear magnetic shielding tensor.

NMR nuclear magnetic shielding anisotropy of linear molecules within the linear response within the elimination of the small component approach.

The influence of the spin-Zeeman (SZ) operator in the evaluation of the spin-orbit effect on the nuclear magnetic shielding tensor in the context of the linear response within the elimination of the small component approach is critically discussed. It is shown that such term yields no contribution to the isotropic nuclear magnetic shielding constant, but it may be of great importance in the determination of individual tensor components, and particularly of the tensor anisotropy. In particular, an interesting relation between the SZ and orbital Zeeman contributions to the spin-orbit effect for the case of linear molecules is shown to hold. Numerical examples for the BrH, IH, and XeF(2) molecules are presented which show that, provided the SZ term is taken into account, results of the individual shielding tensor components and the tensor anisotropy are in good agreement with those obtained by other theoretical methods, and particularly by the Dirac-Hartree-Fock approach.

CLOPPA-IPPP analysis of cooperative effects in h-bonded molecular complexes. 2. Application to the static molecular polarizability tensor.

The cooperative effects on the polarizability tensor are analyzed by means of the IPPP-CLOPPA approach. The decomposition of the polarizability allows classifying these effects in those due to changes in the geometric structure and those that directly involve the transmission mechanisms. This latter contribution admits a further classification, taking into account its electronic origin. As suitable examples, cooperative effects on the polarizability tensor of linear complexes (CNH)(N) and (H(2)O)(N) (N = 2, 3, 4) are discussed.

Relativistic two-component geometric approximation of the electron-positron contribution to magnetic properties in terms of Breit-Pauli spinors.

An alternative approach for the calculation of the electron-positron (e-p) contribution to magnetic properties based on two-component Breit-Pauli spinors is presented. In it, the elimination of the small component scheme is applied to the inverse propagator matrix of e-p pairs. The effect of the positronic manifold is expressed as an operator acting on Breit-Pauli spinors. The operator form thus obtained sums up the relativistic correction as a geometric series and as a result a totally different behavior in the vicinity of a nucleus is obtained as compared to the one of the linear response approximation. This feature has deep influence in numerical values of the e-p contribution to the nuclear magnetic shielding of heavy atoms. Numerical calculations carried out for Kr, Xe, and I show that with this approach, the e-p contributions to this property are in good agreement with those of four-component methods.

CLOPPA-IPPP analysis of cooperative effects in hydrogen-bonded molecular complexes. Application to intermolecular 2hJ(N,C) spin-spin coupling constants in linear (CNH)n complexes.

The cooperative effects on NMR indirect nuclear coupling constants are analyzed by means of the IPPP-CLOPPA approach (where CLOPPA is the Contributions from Localized Orbitals within the Polarization Propagator Approach and IPPP is the Inner Projections of the Polarization Propagator). The decomposition of the J coupling allows one to classify these effects as those due to changes in the geometric structure and those that directly involve the transmission mechanisms. This latter contribution admits a further classification, taking into account its electronic origin. As an example, the cooperative effects on intermolecular 2hJ(N,C) couplings of the linear complexes (CNH)n (n = 2, 3, 4) are discussed.

The sign and magnitude of 2hJ(F,F) and 1hJ(F,H) in FH...FH. A CLOPPA analysis of their distance dependence.

The sign change of the intermolecular (2h)J(F,F) coupling in the (HF)2 dimer as a function of the F-F distance is discussed by means of the CLOPPA method. It is found that it is due to the competition of positive and negative contributions involving the interaction of the sigma lone pair of the acceptor nucleus with vacant molecular orbitals localized in the F-H...F moiety and with other molecular orbitals localized in the donor molecule. The origin of the sign of each contribution is fully determined by analyzing the response of the electronic system to the magnetic perturbation at the acceptor F nucleus. (2h)J(F,F) coupling in the FH...F-, which is positive for all F-F distances, is also analyzed in order to look for the differences with the former case.

CLOPPA-IPPP analysis of electronic mechanisms of intermolecular (1h)J(A,H) and (2h)J(A,D) spin-spin coupling constants in systems with D-H...A hydrogen bonds.

The electronic origin of intermolecular (2h)J(A,D) and (1h)J(A,H) couplings is discussed by means of the CLOPPA-IPPP approach in several model complexes with D-H...A hydrogen bonds. It is found that the origin of these couplings is mainly due to the interaction between the acceptor sigma lone pair and vacant molecular orbitals localized in the D-H...A moiety, regardless of the donor and acceptor nuclei. The problem of the larger absolute value of (2h)J(A,D) compared to (1h)J(A,H) is also addressed.