Constrained Space Orbital Variation Research Articles

Estimating π binding energy of N-heterocyclic carbenes: the role of polarization.

In this work, the tuneability of the π acceptor or donor properties of a set of N-heterocyclic carbenes (NHCs) with a wide spectrum of electronic characteristics is established by means of density functional theory and energy decomposition analysis (EDA) tools. Even though the main orbital interaction contribution to the NHC coordination is the σ donation, a significant contribution of the π interactions to the bond is observed. By means of carefully selected coordination sites, different contributions to the π interactions could be identified and isolated. It includes not only the well known back donation and donation interactions, but also the intrafragment polarization, which has not been considered in previous studies. This can be obtained through the use of the extended transition state method for EDA combined with the natural orbitals for chemical valence and the constrained space orbital variation analysis. The contributions vary with the position of the heteroatoms and the presence of exocyclic substituents; the donation/backdonation π interactions between NHC and the coordination site can range between 2 and 61% of the total π orbital interactions, while the rest is owed to intrafragment polarization. Our results do not only contribute to the understanding of the electronic structure of NHC-based complexes, giving ways to improve their catalytic properties, but also provide comprehension on the modelization methods used to study their donor-acceptor interactions.

Method of Local Increments for the Calculation of Adsorption Energies of Atoms and Small Molecules on Solid Surfaces. 2. CO/MgO(001)

The method of local increments is used in connection with an embedded cluster approach and wave function based quantum chemical ab initio methods to describe the adsorption of a single CO molecule on the MgO(001) surface. The first step in this approach is a conventional Hartree-Fock calculation. The occupied orbitals are then localized by means of the Foster-Boys localization procedure, and the full system is decomposed into several "subunits" that consist of the orbitals localized at the CO molecule and at the Mg and O atoms of the MgO cluster. The correlation energy is expanded into a series of local n-body increments that are evaluated separately and independently. In this way, big savings in computer time can be achieved because (a) the treatment of a large system is replaced with a series of much faster calculations for small subsystems and (b) the big basis sets necessary for describing dispersion effects are only needed for the atoms in the respective subsystem while all other atoms can be treated by medium size Hartree-Fock type basis sets. The coupled electron pair approach, CEPA, an approximate coupled cluster method, is used to calculate the correlation energies of the various subsystems. For the vertical adsorption of CO on top a Mg atom of the MgO(001) surface with the C atom toward Mg, the individual one- and two-body increments are calculated as functions of the CO-MgO separation and a full potential energy curve is constructed from them. A very shallow minimum with an adsorption energy of 0.016 eV at a Mg-C distance of 3.04 Å is found at the Hartree-Fock level, while inclusion of correlation (dispersion) effects shortens the Mg-C distance to 2.59 Å and yields a much larger adsorption energy of 0.124 eV. This is in very good agreement with the best experimental value of 0.14 eV. The basis set superposition error, BSSE, was fully corrected for by the counterpoise method and the bonding mechanism was analyzed at the Hartree-Fock level by means of the constrained space orbital variation, CSOV, analysis.

Importance of backdonation in [M–(CO)]p+ complexes isoelectronic to [Au–(CO)

In this contribution, we study several monocarbonyl-metal complexes in order to unravel the contribution of relativistic effects to the metal-ligand bond length and complexation energy. Using scalar density functional theory (DFT) constrained space orbital variation (CSOV) energy decomposition analysis supplemented by all-electron four-component DFT computations, we describe the dependency of relativistic effects on the orbitals involved in the complexation for the Au(+) isoelectronic series, namely, the fully occupied 5d orbitals and the empty 6s orbitals. We retrieve the well-known sensitivity of gold toward relativity. For platinum and gold, the four-component results illustrate the simultaneous relativistic expansion of the 5d orbitals and the contraction of the 6s orbitals. The consequences of such modifications are evidenced by CSOV computations, which show the importance of both donation and backdonation within such complexes. This peculiar synergy fades away with mercury and thallium for which coordination becomes driven by the accepting 6s orbitals only, which makes the corresponding complexes less sensitive toward the relativistic effects.

On the origin of bonding and vibrational frequency shifts for CO adsorbed on neutral, cationic and anionic gold clusters

We report a detailed analysis of the electronic mechanisms which determine the bond strength and the vibrational frequency of CO molecules adsorbed on neutral or charged gold nanoparticles. To this end we have considered a simple cluster model, Au5COq (q = +1, 0, −1), and decomposed the Au-CO interaction energy into the sum of various contributions according to a Constrained Space Orbital Variation approach. While the adsorption energy is relatively insensitive to the value of q, the C-O stretch frequency, ωe(CO), changes substantially, and allows the use of this molecule as a direct probe of the gold oxidation state. The results show that two major terms contribute to the red or blue shift of ωe(CO) as a function of q: the interaction with the electric field associated to the charged nanoparticle (Stark effect) and the Au → CO Φ back donation. The CO → Au σ donation is about half as important as the Φ back-donation and all other terms are much less important.

Generalization of the Gaussian electrostatic model: extension to arbitrary angular momentum, distributed multipoles, and speedup with reciprocal space methods.

The simulation of biological systems by means of current empirical force fields presents shortcomings due to their lack of accuracy, especially in the description of the nonbonded terms. We have previously introduced a force field based on density fitting termed the Gaussian electrostatic model-0 (GEM-0) J.-P. Piquemal et al. [J. Chem. Phys. 124, 104101 (2006)] that improves the description of the nonbonded interactions. GEM-0 relies on density fitting methodology to reproduce each contribution of the constrained space orbital variation (CSOV) energy decomposition scheme, by expanding the electronic density of the molecule in s-type Gaussian functions centered at specific sites. In the present contribution we extend the Coulomb and exchange components of the force field to auxiliary basis sets of arbitrary angular momentum. Since the basis functions with higher angular momentum have directionality, a reference molecular frame (local frame) formalism is employed for the rotation of the fitted expansion coefficients. In all cases the intermolecular interaction energies are calculated by means of Hermite Gaussian functions using the McMurchie-Davidson [J. Comput. Phys. 26, 218 (1978)] recursion to calculate all the required integrals. Furthermore, the use of Hermite Gaussian functions allows a point multipole decomposition determination at each expansion site. Additionally, the issue of computational speed is investigated by reciprocal space based formalisms which include the particle mesh Ewald (PME) and fast Fourier-Poisson (FFP) methods. Frozen-core (Coulomb and exchange-repulsion) intermolecular interaction results for ten stationary points on the water dimer potential-energy surface, as well as a one-dimensional surface scan for the canonical water dimer, formamide, stacked benzene, and benzene water dimers, are presented. All results show reasonable agreement with the corresponding CSOV calculated reference contributions, around 0.1 and 0.15 kcal/mol error for Coulomb and exchange, respectively. Timing results for single Coulomb energy-force calculations for (H(2)O)(n), n=64, 128, 256, 512, and 1024, in periodic boundary conditions with PME and FFP at two different rms force tolerances are also presented. For the small and intermediate auxiliaries, PME shows faster times than FFP at both accuracies and the advantage of PME widens at higher accuracy, while for the largest auxiliary, the opposite occurs.

Towards accurate solvation dynamics of divalent cations in water using the polarizable amoeba force field: From energetics to structure

Molecular dynamics simulations were performed using a modified amoeba force field to determine hydration and dynamical properties of the divalent cations Ca2+ and Mg2+. The extension of amoeba to divalent cations required the introduction of a cation specific parametrization. To accomplish this, the Thole polarization damping model parametrization was modified based on the ab initio polarization energy computed by a constrained space orbital variation energy decomposition scheme. Excellent agreement has been found with condensed phase experimental results using parameters derived from gas phase ab initio calculations. Additionally, we have observed that the coordination of the calcium cation is influenced by the size of the periodic water box, a recurrent issue in first principles molecular dynamics studies.

The constrained space orbital variation analysis for periodic ab initio calculations

The constrained space orbital variation (CSOV) method for the analysis of the interaction energy has been implemented in the periodic ab initio CRYSTAL03 code. The method allows for the partition of the energy of two interacting chemical entities, represented in turn by periodic models, into contributions which account for electrostatic effects, mutual polarization and charge transfer. The implementation permits one to carry out the analysis both at the Hartree-Fock and density functional theory levels, where in the latter the most popular exchange-correlation functionals can be used. As an illustrating example, the analysis of the interaction between CO and the MgO (001) surface has been considered. As expected by the almost fully ionic character of the support, our periodic CSOV results, in general agree with those previously obtained using the embedded cluster approach, showing the reliability of the present implementation.

Towards a force field based on density fitting

Total intermolecular interaction energies are determined with a first version of the Gaussian electrostatic model (GEM-0), a force field based on a density fitting approach using s-type Gaussian functions. The total interaction energy is computed in the spirit of the sum of interacting fragment ab initio (SIBFA) force field by separately evaluating each one of its components: electrostatic (Coulomb), exchange repulsion, polarization, and charge transfer intermolecular interaction energies, in order to reproduce reference constrained space orbital variation (CSOV) energy decomposition calculations at the B3LYP/aug-cc-pVTZ level. The use of an auxiliary basis set restricted to spherical Gaussian functions facilitates the rotation of the fitted densities of rigid fragments and enables a fast and accurate density fitting evaluation of Coulomb and exchange-repulsion energy, the latter using the overlap model introduced by Wheatley and Price [Mol. Phys. 69, 50718 (1990)]. The SIBFA energy scheme for polarization and charge transfer has been implemented using the electric fields and electrostatic potentials generated by the fitted densities. GEM-0 has been tested on ten stationary points of the water dimer potential energy surface and on three water clusters (n = 16,20,64). The results show very good agreement with density functional theory calculations, reproducing the individual CSOV energy contributions for a given interaction as well as the B3LYP total interaction energies with errors below kBT at room temperature. Preliminary results for Coulomb and exchange-repulsion energies of metal cation complexes and coupled cluster singles doubles electron densities are discussed.

Revisiting the geometry of nd10 (n+1)s0 [M(H2O)]p+ complexes using four‐component relativistic DFT calculations and scalar relativistic correlated CSOV energy decompositions (Mp+ = Cu+, Zn2+, Ag+, Cd2+, Au+, Hg2+)

Hartree-Fock and DFT (B3LYP) nonrelativistic (scalar relativistic pseudopotentials for the metallic cation) and relativistic (molecular four-component approach coupled to an all-electron basis set) calculations are performed on a series of six nd10 (n+1)s0 [M(H2O)]p+ complexes to investigate their geometry, either planar C2v or nonplanar C(s). These complexes are, formally, entities originating from the complexation of a water molecule to a metallic cation: in the present study, no internal reorganization has been found, which ensures that the complexes can be regarded as a water molecule interacting with a metallic cation. For [Au(H2O)]+ and [Hg(H2O)]2+, it is observed that both electronic correlation and relativistic effects are required to recover the C(s) structures predicted by the four-component relativistic all-electron DFT calculations. However, including the zero-point energy corrections makes these shallow C(s) minima vanish and the systems become floppy. In all other systems, namely [Cu(H2O)]+, [Zn(H2O)]2+, [Ag(H2O)]+, and [Cd(H2O)]2+, all calculations predict a C2v geometry arising from especially flat potential energy surfaces related to the out-of-plane wagging vibration mode. In all cases, our computations point to the quasi-perfect transferability of the atomic pseudopotentials considered toward the molecular species investigated. A rationalization of the shape of the wagging potential energy surfaces (i.e., single well vs. double well) is proposed based on the Constrained Space Orbital Variation decompositions of the complexation energies. Any way of stabilizing the lowest unoccupied orbital of the metallic cation is expected to favor charge-transfer (from the highest occupied orbital(s) of the water ligand), covalence, and, consequently, C(s) structures. The CSOV complexation energy decompositions unambiguously reveal that such stabilizations are achieved by means of relativistic effects for [Au(H2O)]+, and, to a lesser extent, for [Hg(H2O)]2+. Such analyses allow to numerically quantify the rule of thumb known for Au+ which, once again, appears as a better archetype of a relativistic cation than Hg2+. This observation is reinforced due to the especially high contribution of the nonadditive correlation/relativity terms to the total complexation energy of [Au(H2O)]+.

A CSOV study of the difference between HF and DFT intermolecular interaction energy values: The importance of the charge transfer contribution

Intermolecular interaction energy decompositions using the Constrained Space Orbital Variation (CSOV) method are carried out at the Hartree-Fock level on the one hand and using DFT with usual GGA functionals on the other for a number of model complexes to analyze the role of electron correlation in the intermolecular stabilization energy. In addition to the overall stabilization, the results provide information on the variation, with respect to the computational level, of the different contributions to the interaction energy. The complexes studied are the water linear dimer, the N-methylformamide dimer, the nucleic acid base pairs, the benzene-methane and benzene-N2 van der Waals complexes, [Cu+ -(ImH)3]2, where "ImH" stands for the Imidazole ligand, and ImH-Zn++. The variation of the frozen core energy (the sum of the intermolecular electrostatic energy and the Pauli repulsion energy) calculated from the unperturbed orbitals of the interacting entities indicates that the intramolecular correlation contributions can be stabilizing as well as destabilizing, and that general trends can be derived from the results obtained using usual density functionals. The most important difference between the values obtained from HF and DFT computations concerns the charge transfer contribution, which, in most cases, undergoes the largest increase. The physical meaning of these results is discussed. The present work gives reference calculations that might be used to parametrize new correlated molecular mechanics potentials.

Theoretical study of the chemisorption of CO on bimetallic RhCu surfaces and nanoparticles

The CO interaction with bimetallic RhCu surface models representing several compositions has been studied by first principles density functional theory calculations. The analysis of the bare bimetallic clusters Rh(4s) and Cu(3s) core-level binding energies indicates that is not possible to extract information about the oxidation state of the alloy components. The present calculations predict that CO does always sit on top sites, the influence of the alloy composition on the equilibrium geometry and vibrational frequency of CO chemisorbed at a given Rh or Cu site being very small. However, there is a large difference in the structural properties corresponding to CO chemisorption above either Rh or Cu. Therefore, the absolute value of the vibrational frequency of chemisorbed CO does not permit to extract any information about the alloy composition but afford to assign the chemisorption site. Finally, the CO adsorption energy does not follow a monotonic trend with composition. The use of the Constrained Space Orbital Variation analysis permits one to firmly establish that the difference in adsorption energy for different compositions cannot be explained through differences in the σ-donation and π-backdonation mechanisms.

An ab initio study of the adsorption of CO on a Zn4O4 cluster with wurtzite-like structure

The adsorption of CO on a free Zn 4 O 4 cluster with a wurtzite-like structure is studied by means of quantum-chemical ab initio methods at the Hartree–Fock level and with inclusion of electron correlation effects. The Zn 4 O 4 cluster contains twofold and threefold coordinated Zn and O atoms and can therefore serve as a model substrate for the adsorption of CO on different adsorption sites at terraces, step edges, and corners of various ZnO surface planes. The calculations show that CO binds most strongly to the twofold and the threefold coordinated Zn atoms; the calculated binding energies are 0.52 and 0.33 eV, respectively. The preferred adsorption geometries are “end-on”, with the C atom towards Zn and linear Zn–C–O configurations. The bonding of CO to the oxygen atoms of the Zn 4 O 4 cluster, on the other hand, is very weak; the binding energies are less than 0.1 eV and the “side-on” adsorption geometries are more favourable. The mechanism of the bonding between CO and the Zn 4 O 4 cluster as well as the differences in the bonding properties of the various adsorption sites are explained by means of a constrained space orbital variation (CSOV) analysis. A comparison with the bonding of CO to an isolated Zn 2+ ion is also included.

The Effect of Pyrimidine Bases on the Hole-Transfer Coupling in DNA†

Electron hole migration along the π-stack of DNA is mediated by the electronic coupling between the purines (guanine, adenine) of adjacent Watson-Crick pairs (WCPs). We have investigated the effect of the complementary pyrimidines (cytosine, thymine) on this coupling using the two-state model of electron transfer where, at the Hartree-Fock level, the hole-transfer coupling can be described by the energy splitting of the HOMO and HOMO-1 molecular orbitals of WCP dimers. Using the constrained space orbital variation method, we considered the effects of exchange, polarization, and electrostatic interactions (including hydrogen bonds of a WCP) of pyrimidines on the purine-purine electronic couplings. We also performed a detailed analysis of HOMO and HOMO-1 of the duplexes. We conclude that the effect of pyrimidines on the electronic coupling of neighboring purines cannot be described via perturbation theory nor simply reduced to an electrostatic polarization of the purines. However, the combination of exchange (frozen core) and electrostatic effects reproduces the coupling matrix elements between adjacent WCPs rather well. Analysis of the spatial overlap between orbital fragments of HOMO and HOMO-1 suggests that hydrogen bonds between complementary bases play a crucial role in the electronic coupling of neighboring WCPs.

Similarities and differences in the Hartree–Fock and density-functional description of the chemisorption bond

The similarities and differences between the ab initio Hartree–Fock (HF) and density-functional theory (DFT) descriptions of the chemisorption bond have been explored by applying the constrained space orbital variation (CSOV) method to obtain and analyze the HF and DFT total energies of cluster models representing the interaction of CO and NH 3 on Cu(100) and Pt(111). The qualitative picture of the chemisorption bond arising from ab initio HF and DFT quantum-chemical approaches is essentially the same; the relative importance of the different mechanisms remains unchanged. The main quantitative effect of electronic correlation is to increase the inter-unit charge transfer but substrate polarization is also affected. A very important consequence emerges: it is not necessary to revise the physical mechanisms that have been previously proposed from the ab initio Hartree–Fock cluster model approach.

Ab initio cluster model approach to the chemisorption of NH 3 on Pt(111)

The chemisorption of NH 3 on Pt(111) has been studied using several Pt 10 clusters that model different adsorption sites of the Pt(111) surface. Ab initio methods have been used to obtain a theoretical estimate of several spectroscopic features that can be directly compared to experiment. The comparison includes the variation of the difference between the 3a 1 and 1e levels, the vibrational frequency shifts and the order of stability on different surface sites. Chemisorption at the on-top site is predicted to be favoured, the calculated interaction energy appears to be quite close to the experimental estimate, and it is suggested that NH 3 chemisorbs molecularly, in an N-down orientation, with an equilibrium geometry representing a small distortion from the gas-phase molecular geometry and no azimuthal preference, in good agreement with ESDIAD experiments. Constrained space orbital variation (CSOV) analysis of the interaction has also been performed using a Hartree–Fock wave function. This analysis shows that the leading bonding mechanisms are substrate polarisation and charge transfer from ammonia to the surface.

Ab Initio Cluster Model Study of the Chemisorption of CO on Low-Index Platinum Surfaces

A systematic theoretical study of the adsorption of CO on the Pt{100}, Pt{110}, and Pt{111} surfaces is presented. The calculated equilibrium geometries and vibrational frequencies have been found to be rather independent of the cluster model chosen to represent the surface. However, calculated interaction energies are found to be very sensitive to the surface cluster model. The analysis of the chemisorption bond has been carried out by means of the constrained space orbital variation, CSOV, and of projection operator techniques. These analysis reveal that the bonding interactions are dominated by the π-back-donation although σ-donation plays a significant role. It is also clearly shown that all bonding mechanisms, other than Pauli repulsion, but specially π-back-donation, contribute to the observed red shift. However, the π-back-donation contribution to the red shift is very similar for CO on different sites. Hence, π-back-donation cannot be the mechanism responsible for the observed difference for the CO vibrational frequency on on-top and bridge sites. The CSOV decomposition reveals that the leading term contributing to this difference in vibrational frequency of chemisorbed CO is the initial Pauli repulsion or “wall effect”; this is a new, important and unexpected conclusion.

Ab initio cluster model comparative study of atomic oxygen and sulfur chemisorption on Pt(1 1 1) surface: relevance to heterogeneous catalysis

Chemisorption of atomic oxygen and sulfur on Pt(1 1 1) has been studied by means of the ab initio cluster model approach. For both adsorbates, we consider chemisorption on the threefold open site of the Pt(1 1 1) surface which is represented by a Pt 25 cluster model having 12 atoms in the first layer, six in the second and seven in the third. The ab initio molecular orbital Hartree–Fock method and explicitly correlated wavefunctions are used to obtain a reliable estimate of the interaction energy, vibrational frequency for the normal mode of the adsorbate above the surface, and the equilibrium geometry. The chemisorption bond is analyzed using different theoretical techniques including the constrained space orbital variation, CSOV, method, the analysis of dipole moment curves and the use of projection operators. The influence of electronic correlation effects is analyzed using multireference configuration interaction, MRCI, techniques and also, by using post-Hartree–Fock correlation functionals.

Five-coordinate platinum (II) alkyne complexes: synthesis, ab initio calculations and crystal and molecular structure of [PtI 2(Me 2phen) η2PhC CPh]·CHCl 3

[PtX 2(Me 2phen)] complexes ( 1)Me 2phen = 2,9-dimethyl-1,10-phenanthroline; X = Br, Iy; I, Iz), having a highly distorted square-planar geometry, undergo alkynes uptake in chlorinated solvents to give the five-coordinate trigonal-bipyramidal species [PtX 2(Me 2phen)-(alkyne)] ( 2) (alkyne = ethyne, a; propyne, b; 2-butyne, c; 1-pentyne, d; phenylacetylene, e; methylphenylacetylene, f; diphenylacetylene, g). The equilibrium constants for formation ( K 1) decrease in the order (1>Br) already observed for the analogous reaction with alkenes to give the [PtX 2(Me 2phen)(alkene)] complexes ( 3) (1>Br>Cl) although the differences in K 1 values are much greater in the case of alkynes. In contrast, the number and bulk of substituents at the metal-coordinated carbons have practically no effect on the K 1 of the alkyne complexes, while they greatly reduce the stability of the five-coordinate olefin compounds. Finally, the Δ G of activation for rotation around the Ptalkyne bond calculated by variable temperature 1H NMR spectroscopy (71.5 ± 0.8 kJ mol −1 for 2zb) is considerably smaller than the value found for the corresponding five-coordinate propene complex (86.2 ± 0.8 kJ mol −1). The crystal and molecular structure of the chloroform solvate of [PtI 2(Me 2phen( η 3-PhCCPh)] ( 2zg·CHCl 3) has been determined by X-ray diffraction methods. In the crystals two crystallographically independent complexes and chloroform molecules are present: space group Cmc2, a = 14.367(3), b = 15.529(3), c = 26.959(5)Å, Z = 8, R = 0.0444. The average bending back of the phenyl groups with respect to the CC triple bond axis is 25(1)°. Spectroscopic and crystal data show that the conformation of the coordinated alkynes in these five-coordinate complexes is intermediate between those found for the trigonal Pt(0) and the square-planar Pt(II) alkyne complexes. Ab initio Hartfree-Fock calculations and constrained space orbital variation analyses have shown that in the five-coordinate species ethylene exhibits better σ-donating and π-accepting properties than acetylene.

Strain Effects in Protonated Carbonyl Compounds. An Experimental and ab Initio Treatment of Acyclic Carboxamides and Ketones.

Strain effects have been quantitatively evaluated for a set of 22 compounds including ketones (R(2)CO), carboxamides (RCONH(2)), and N,N-dimethylcarboxamides (RCONMe(2)), where R = Me, Et, i-Pr, t-Bu, 1-adamantyl (1-Ad), in their neutral and protonated forms. To this end, use was made of the gas-phase proton affinities and standard enthalpies of formation of these compounds in the gas phase, as determined by Fourier transform ion cyclotron resonance mass spectrometry (FT ICR) and thermochemical techniques, respectively. The structures of 1-AdCOMe and (1-Ad)(2)CO were determined by X-ray crystallography. Quantum-mechanical calculations, at levels ranging from AM1 to MP2/6-311+G(d,p)//6-31G(d), were performed on the various neutral and protonated species. Constrained space orbital variation (CSOV) calculations were carried out on selected protonated species to further assess the contributions of the various stabilizing factors. Taking neutral and protonated methyl ketones as references, we constructed isodesmic reactions that provided, seemingly for the first time, quantitative measures of strain in the protonated species. A combination of these data with the results of theoretical calculations (which also included several "computational experiments") lead to a unified, conceptually satisfactory, quantitative description of these effects and their physical link to structural properties of the neutral and protonated species.

Effect of surface relaxation and rumpling on the vibrational spectrum of NO adsorbed on a Cu 2O surface

We report theoretical results for the bonding of NO to a singly coordinated Cu + cation of the Cu 2O(111) surface. The effect of the surface relaxation and of the rumpling of coordinatively unsaturated cations on the energy and bonding nature of the interaction and on the vibrational spectrum has been analyzed by using the constrained space orbital variation, CSOV, technique. We found that the bonding is always electrostatic in nature with negligible electron transfer between adsorbate and surface. Surface relaxation phenomena alter the balance among the significant contributions (electrostatic, Pauli repulsion and NO polarization) to the NO stretch frequency, suggesting that these effects, rather than charge transfer ones, may be in the physical origin of the red shift experimentally observed on NO Cu 2O systems.