Calculation of electric dipole hypershieldings at the nuclei in the Hellmann-Feynman approximation

The third-rank electric hypershieldings at the nuclei of four small molecules have been evaluated at the Hartree-Fock level of theory in the Hellmann-Feynman approximation. The nuclear electric hypershieldings are closely related to molecular vibrational absorption intensities and a generalization of the atomic polar tensors (expanded in powers of the electric field strength) is proposed to rationalize these intensities. It is shown that the sum rules for rototranslational invariance and the constraints imposed by the virial theorem provide useful criteria for basis-set completeness and for near Hartree-Fock quality of nuclear shieldings and hypershieldings evaluated in the Hellmann-Feynman approximation. Twelve basis sets of different size and quality have been employed for the water molecule in an extended numerical test on the practicality of the proposed scheme. The best results are obtained with the R12 and R12+ basis sets, designed for the calculation of electronic energies by the explicitly correlated R12 method. The R12 basis set is subsequently used to investigate three other molecules, CO, N2, and NH3, verifying that the R12 basis consistently performs very well.     

Calculation of the electric hypershielding at the nuclei of molecules in a strong magnetic field

The third-rank electric hypershielding at the nuclei of 14 small molecules has been evaluated at the Hartree-Fock level of accuracy, by a pointwise procedure for the geometrical derivatives of magnetic susceptibilities and by a straightforward use of its definition within the Rayleigh-Schrodinger perturbation theory. The connection between these two quantities is provided by the Hellmann-Feynman theorem. The magnetically induced hypershielding at the nuclei accounts for distortion of molecular geometry caused by strong magnetic fields and for related changes of magnetic susceptibility. In homonuclear diatomics H(2), N(2), and F(2), a field along the bond direction squeezes the electron cloud toward the center, determining shorter but stronger bond. It is shown that constraints for rotational and translational invariances and hypervirial theorems provide a natural criterion for Hartree-Fock quality of computed nuclear electric hypershielding.     

Forces on the Nuclei of a Molecule in Optical Fields

The electric Lorentz forces acting upon the nuclei of a vibrating molecule cause variations of dynamical regime and determine the intensity of the absorbed radiation. These forces, depending on the local electric field, can be evaluated by frequency-dependent electric and electromagnetic shielding and hypershielding tensors at the nuclei. A general expression from time-dependent perturbation theory is all that one needs to rationalize the molecular response by predicting the effective electric field at the nuclei of a molecule perturbed by an external monochromatic wave. The electric and electromagnetic hypershieldings are connected with the geometrical derivatives of the frequency-dependent dipole polarisability and of the optical rotatory power, respectively. Intensities in Raman spectroscopy and in vibrational Raman optical activity, usually interpreted in terms of these derivatives, can also be discussed via nuclear electromagnetic hypershieldings. Conditions for translational and rotational invariance can be expressed via sum rules for the dynamic hypershieldings.Article submitted for the issue in honour of J. P. Malrieu     

How (77)Se NMR chemical shifts originate from pre-alpha, alpha, beta, and gamma effects: interpretation based on molecular orbital theory

Plain rules founded in a theoretical background are presented that can be used to determine the structure of selenium compounds on the basis of delta(Se) data and to predict delta(Se) data from a given structure with satisfactory accuracy. As a first step to establish such rules, the origin of delta(Se) is elucidated on the basis of MO theory. The Se(2-) ion was chosen as the standard for the analysis. The concept of the pre-alpha effect is proposed, which is defined as the downfield shift due to protonation of a lone-pair orbital of Se. The pre-alpha effect of two protons in H(2)Se is explained by the generation of double sigma(Se--H) and sigma*(Se--H) through protonation of the spherical Se(2-) ion. The orbitals, together with n(p)(Se), result in effective transitions for the pre-alpha effect. The alpha effect is the downfield shift caused by the replacement of Se--H by Se--Me. The extension of HOMO-2 [4p(y)(Se)], HOMO-1 [4p(x)(Se)], and HOMO [4p(z)(Se)] over the whole Me(2)Se molecule is mainly responsible for the alpha effect. The beta effect originates not from the occupied-to-unoccupied (psi(i)-->psi(a)) transitions but from the occupied-to-occupied (psi(i)-->psi(j)) transitions. Although psi(i)-->psi(j) transitions contribute to upfield shifts in Me(2)Se, the magnitudes become smaller as the methyl protons are substituted by Me groups one after another. The gamma effect of upfield shifts is also analyzed, although complex. The effect of p(Se)-pi(C==C) conjugation is analyzed in relation to the orientational effect. Contributions from each MO (psi(i)) and each psi(i)-->psi(a) transition are evaluated separately, by using a utility program derived from the Gaussian 03 program suite (NMRANAL-NH03G). The treatment enables us to visualize and understand the origin of (77)Se NMR chemical shifts.     

Multiresolution quantum chemistry in multiwavelet bases: Analytic derivatives for Hartree-Fock and density functional theory

An efficient and accurate analytic gradient method is presented for Hartree-Fock and density functional calculations using multiresolution analysis in multiwavelet bases. The derivative is efficiently computed as an inner product between compressed forms of the density and the differentiated nuclear potential through the Hellmann-Feynman theorem. A smoothed nuclear potential is directly differentiated, and the smoothing parameter required for a given accuracy is empirically determined from calculations on six homonuclear diatomic molecules. The derivatives of N2 molecule are shown using multiresolution calculation for various accuracies with comparison to correlation consistent Gaussian-type basis sets. The optimized geometries of several molecules are presented using Hartree-Fock and density functional theory. A highly precise Hartree-Fock optimization for the H2O molecule produced six digits for the geometric parameters.     

Calculation of the force constants of molecules by means of the Hellmann-Feynman theorem

The force constants of the molecules LiH, BH, and H₂O have been calculated by means of the Hellmann-Feynman theorem on the basis of the Hartree-Fock-Roothan wave functions in the one-determinant approximation. The results obtained have been compared with the results previously calculated on the basis of the same functions by means of the virial theorem.     

Probing internal electric fields of π- and σ-hole bonds

π- and σ-holes are nonnuclear molecular regions of positive electric potential, which make non-covalent interactions with negative sites, for example, lone pairs of molecules containing nitrogen or oxygen, the so called π- and σ-hole bonds. We investigate these bonds locally using a probe programmed as a virtual molecule. Unlike the hydrogen bond, electric fields are detected having strengths that are different from the sum of the separated parts, meaning that molecular electrostatic potential surfaces analysis of the different parts are not enough to analyze the bonds. Based on an application of the Hellmann-Feynman theorem, which states that intermolecular bonds are fully described by Coulombian interactions (electrostatic plus polarization), we connect the electric field strength with the bond strength measured in experiments, so that it can be considered as a quantifier for the bonds.     

Analytic derivative couplings between configuration-interaction-singles states with built-in electron-translation factors for translational invariance

We present a method for analytically calculating the derivative couplings between a pair of configuration-interaction-singles (CIS) excited states obtained in an atom-centered basis. Our theory is exact and has been derived using two completely independent approaches: one inspired by the Hellmann-Feynman theorem and the other following from direct differentiation. (The former is new, while the latter is in the spirit of existing approaches in the literature.) Our expression for the derivative couplings incorporates all Pulay effects associated with the use of an atom-centered basis, and the computational cost is minimal, roughly comparable to that of a single CIS energy gradient. We have validated our method against CIS finite-difference results and have applied it to the lowest lying excited states of naphthalene; we find that naphthalene derivative couplings include Pulay contributions sufficient to have a qualitative effect. Going beyond standard problems in analytic gradient theory, we have also constructed a correction, based on perturbative electron-translation factors, for including electronic momentum and eliminating spurious components of the derivative couplings that break translational symmetry. This correction is general and can be applied to any level of electronic structure theory.     

Floating functions satisfying the hellmann—feynman theorem: Single floating scheme

The electrostatic calculation for molecules using approximated variational wave functions leads to well known difficulties connected with the application of the Hellmann-Feynman (H? F) theorem. This is due to the basis set inadequacies in the underlying calculations. This defect can easily be remedied by floating functions, whose centers are optimized in space. We can keep almost everything of the traditional wave function with a nuclear-fixed basis set, but we apply single floating to ensure the H? F theorem. Then, one can obtain a wave function obeying the H? F theorem. This provides a great conceptual simplification and may lead to practical advantages. The single floating scheme, which retains one expansion center per nucleus, is successfully applied to a series of small molecules using SCF and CASSCF wave functions with sufficiently polarized basis sets.     

Generation of basis sets with high degree of fulfillment of the Hellmann-Feynman theorem

A direct relationship is established between the degree of fulfillment of the Hellman-Feynman (electrostatic) theorem, measured as the difference between energy derivatives and electrostatic forces, and the stability of the basis set, measured from the indices that characterize the distance of the space generated by the basis functions to the space of their derivatives with respect to the nuclear coordinates. On the basis of this relationship, a criterion for obtaining basis sets of moderate size with a high degree of fulfillment of the theorem is proposed. As an illustrative application, previously reported Slater basis sets are extended by using this criterion. The resulting augmented basis sets are tested on several molecules finding that the differences between energy gradient and electrostatic forces are reduced by at least one order of magnitude.     

Bonding and (hyper)polarizability in the sodium dimer

We report a conventional ab initio and density functional theory study of the polarizability (alpha(alphabeta)/e(2)a(0) (2)E(h) (-1)) and hyperpolarizability (gamma(alphabetagammadelta)/e(4)a(0) (4)E(h) (-3)) of the sodium dimer. A large [18s14p9d2f1g] basis set is thought to yield near-Hartree-Fock values for both properties: alpha=272.28, Deltaalpha=127.22 and gamma=2157.6 x 10(3) at R(e)=3.078 87 A. Electron correlation has a remarkable effect on the Cartesian components of gamma(alphabetagammadelta). Our best value for the mean is gamma=1460.1 x 10(3). The (hyper)polarizability shows very strong bond-length dependence. The effect is drastically different for the longitudinal and transverse components of the hyperpolarizability. The following first derivatives were extracted from high-level coupled cluster calculations: (dalpha/dR)(e)=54.1, (dDeltaalpha/dR)(e)=88.1e(2)a(0)E(h) (-1), and (dgamma/dR)(e)=210 x 10(3)e(4)a(0) (3)E(h) (-3). We associate the (hyper)polarizability to bonding effects between the two sodium atoms by introducing the differential property per atom Q(diff)/2 identical with (Q[Na(2)(X (1)Sigma(g) (+))]/2-Q[Na((2)S)]). The differential (hyper)polarizability per atom is predicted to be strongly negative for the dimer at R(e), as [alpha(Na(2))/2-alpha(Na)]=-33.8 and [gamma(Na(2))/2-gamma(Na)]=-226.3 x 10(3). The properties calculated with the widely used B3LYP and B3PW91 density functional methods differ significantly. The B3PW91 results are in reasonable agreement with the conventional ab initio values. Last, we observe that low-level ab initio and density functional theory methods underestimate the dipole polarizability anisotropy. Experimental data on this important property are highly desirable.     

Electric quadrupole contribution to the nonresonant background of sum frequency generation at air/liquid interfaces

We study an electric quadrupole contribution to sum frequency generation (SFG) at air∕liquid interfaces in an electronically and vibrationally nonresonant condition. Heterodyne-detected electronic sum frequency generation spectroscopy of air∕liquid interfaces reveals that nonresonant χ((2)) (second-order nonlinear susceptibility) has a negative sign and nearly the same value for all eight liquids studied. This result is rationalized on the basis of the theoretical expressions of χ((2)) with an electric quadrupole contribution taken into account. It is concluded that the nonresonant background of SFG is predominantly due to interfacial nonlinear polarization having a quadrupole contribution. Although this nonlinear polarization is localized at the interface, it depends on quadrupolar χ((2)) in the bulk as well as that at the interface. It means that the sign of nonresonant χ((2)) bears no relation to the "up" versus "down" alignment of interfacial molecules, because nonresonant χ((2)) has a quadrupolar origin.     

Analytical calculation of atomic and molecular electrostatic potentials from the Poisson equation

An analytic formulation is given for the total potential in atomic and molecular systems, based on the electrostatic approach from the Hellmann-Feynman theorem. The potential function is obtained from the analytic solution of the Poisson equation using charge densities expressed as a superposition of gaussian functions. The method is independent of the specific LCAO approximation used for the calculation of the charge distribution function. The calculation of the potential and its derivatives to a rapid algorithm form, which can be used for the evaluation of various electronic properties and the treatment of experimental situation, even for large molecular systems.     

Self-assembly of N2-modified guanosine derivatives: formation of discrete G-octamers

In the presence of Na(+) ions, two N(2)-modified guanosine derivatives, N(2)-(4-n-butylphenyl)-2',3',5'-O-triacetylguanosine (G1) and N(2)-(4-pyrenylphenyl)-2',3',5'-O-triacetylguanosine (G2), are found to self-associate into discrete octamers that contain two G-quartets and a central ion. In each octamer, all eight guanosine molecules are in a syn conformation and the two G-quartets are stacked in a tail-to-tail fashion. On the basis of NMR spectroscopic evidence, we hypothesize that the pi-pi-stacking interaction between the N(2)-side arms (phenyl in G1 and pyrenyl in G2) can considerably stabilize the octamer structure. For G1, we have used NMR spectroscopic saturation-transfer experiments to monitor the kinetic ligand exchange process between monomers and octamers in CD(3)CN. The results show that the activation energy (E(a)) of the ligand exchange process is 31 +/-5 kJ mol(-1). An Eyring analysis of the saturation transfer data yields the enthalpy and entropy of activation for the transition state: DeltaH(not =)=29 +/-5 kJ mol(-1) and DeltaS(not =)=-151 +/-10 J mol(-1) K(-1). These results are consistent with an associative mechanism for ligand exchange.     

Geometry optimization inab initio SCF calculations

The floating orbital geometry optimization (FOGO) described previously [1, 2] for atoms without polarized inner-shell electrons, is extended to the general case. Instead of the Hellmann-Feynman force a special gradient is calculated analytically and utilized in a variable metric procedure simultaneously with the ordinary energy gradient. Test calculations on a sample of 12 molecules were performed to check the efficiency of the method. The geometries obtained are better than those obtained with the corresponding double-zeta basis set. The most striking results, however, are excellent dipole moments.     

Electric Potential (psi(delta) and psi(d)) at Outer Helmholtz Plane and Midplane on the Clay Colloid Surface with Overlapping Flat Double Layers

An anion negative adsorption equation in the condensed colloidal suspension with overlapping flat double layers was derived according to Gouy-Chapman theory. The electric potential at the outer Helmholtz plane (OHP), psi(delta), and the electric potential at the midplane, psi(d), were numerically solved by computer using the anion negative adsorption equation on the basis of experiments. The results showed that psi(delta) and psi(d) increase with the decrease of the distance between two clay plates, lambda, at first in the given electrolyte concentration. When lambda is smaller than 50-70 ?, psi(d) remains almost unchanged while psi(delta) declines remarkably with the further decrease of lambda. The change of psi(d)/psi(delta) with lambda can explain and manifest overlapping degree of flat double layers more appropriately than psi(d) in previous works. Due to compression of the flat double layer on the clay colloid surface at increasing electrolyte concentration, the magnitude of the electrical potentials at OHP and midplane is considerably reduced at a given lambda. Copyright 2000 Academic Press.     

Hydrogen bonding in phenol, water, and phenol-water clusters

Structure, stability, and hydrogen-bonding interaction in phenol, water, and phenol-water clusters have been investigated using ab initio and density functional theoretical (DFT) methods and using various topological features of electron density. Calculated interaction energies at MP2/6-31G level for clusters with similar hydrogen-bonding pattern reveal that intermolecular interaction in phenol clusters is slightly stronger than in water clusters. However, fusion of phenol and water clusters leads to stability that is akin to that of H(2)O clusters. The presence of hydrogen bond critical points (HBCP) and the values of rho(r(c)) and nabla(2)rho(r(c)) at the HBCPs provide an insight into the nature of closed shell interaction in hydrogen-bonded clusters. It is shown that the calculated values of total rho(r(c)) and nabla(2)rho(r(c)) of all the clusters vary linearly with the interaction energy.     

Aurophilicity versus mercurophilicity: impact of d10-d10 metallophilic interactions on the structure of metal-rich supramolecular assemblies

Treatment of U-shaped, binuclear Cu(I) complexes 1,1' (1, counterion: BF(4)(-); 1', counterion: PF(6)(-)) with metal cyanide linear linkers K[Au(CN)(2)] (3) and Hg(CN)(2) (4) lead to formation of new supramolecular assemblies 5,5' and 6,6', respectively, in good yield. These derivatives have been characterized by NMR spectroscopy, IR, and X-ray diffraction studies. Derivative 5,5' are supramolecular metallacycles in which intramolecular aurophilic interactions between the Au(I) metal centers of the linkers are observed. Derivative 5 crystallizes as a single solid phase, whereas derivative 5' is characterized in the solid state as four different pseudo-polymorphs (5'a-d). Notably in the case of phase 5'd, a dimer of supramolecular metallacycles bounded by intermolecular aurophilic interactions is formed. Conversely, derivatives 6,6' present large structural diversity depending on the nature of the counterion. Derivative 6 is a supramolecular rectangle in which the Hg(II)-Hg(II) metal distance suggests mercurophilic interaction, whereas 6' crystallizes as two different pseudo-polymorphs 6'a,b, that is, a one-dimensional coordination polymer and one oligomer with no short Hg(II)-Hg(II) metal contacts, respectively. In derivatives 6,6', short contacts between the Hg(II) metal centers and fluorine atoms of the counterions are also observed, which may explain the counterion structural dependence of these supramolecular assemblies based on Hg(II) metal cyanide linker. Comparison of the different solid-state structures characterized highlights the importance of weak secondary interactions between the linkers for the formation supramolecular metallacycles from molecular clips 1,1' and suggests the range of energies required for these interactions to form metallacycles and to induce self-aggregation.     

pi Molecular Orbital Crossing a(2)(chi)/b(1)(psi) in 1,10-Phenanthroline Derivatives. Ab Initio Calculations and EPR/ENDOR Studies of the 4,7-Diaza-1,10-phenanthroline Radical Anion and Its M(CO)(4) Complexes (M = Cr, Mo, W)

Ab initio, semiempirical, and HMO perturbation calculations were employed to assess the relative positioning of the closely situated low-lying unoccupied pi MOs a(2)(chi) and b(1)(psi) in 1,10-phenanthroline (phen) and its 3,4,7,8-tetramethyl (tmphen) and four symmetrical diaza derivatives (n,m-dap). Compared to a(2)(chi), the b(1)(psi) pi MO is distinguished by markedly higher MO coefficients at the chelating nitrogen pi centers in 1,10-positions; eventually, a higher Coulomb integral value at those positions will thus always favor the lowering of b(1) beyond a(2). Using the Coulomb integral parameter at the chelating 1,10-nitrogen pi centers as the HMO perturbation variable, the crossing of both energy levels in terms of decreasing preference for the a(2)(chi) over the b(1)(psi) orbital as the lowest unoccupied MO follows the sequence 5,6-dap > 2,9-dap > 4,7-dap > phen > 3,8-dap. The calculations reveal a(2)(chi) as the LUMO in 5,6-dap for all reasonable perturbation parameters, in agreement with previous observations for ruthenium(II) complexes which reveal a discrepancy between the lowest-lying "redox pi orbital" (a(2)) and the "optical pi MO" (b(1)) to which the most intense low-energy MLCT transition occurs. According to the HMO calculations, the situation is more ambiguous for the 4,7-dap analogue, yet EPR/ENDOR studies clearly show that the one-electron-reduced ligand and its tetracarbonylmetal(0) complexes (Cr, Mo, W) have the b(1)(psi) orbital singly occupied. Only ab initio calculations with double-zeta basis and inclusion of d polarization functions reproduced correctly the experimentally observed orbital ordering for tmphen (a(2) < b(1)) and for phen and 4,7-dap (b(1) < a(2)).     

Ab initio molecular orbital study on the G-selectivity of GGG triplet in copper(I)-mediated one-electron oxidation

The G-selectivity for Cu(I)-mediated one-electron oxidation of 5'-TG(1)G(2)G(3)-3' and 5'-CG(1)G(2)G(3)-3' has been examined by ab initio molecular orbital calculations. It was confirmed that G(1) is selectively damaged by Cu(I) ion for both 5'-TG(1)G(2)G(3)-3' and 5'-CG(1)G(2)G(3)-3', being good agreement with experimental results. The Cu(I)-mediated G(1)-selectivity is primarily due to the stability of the Cu(I)-coordinated complex, [-XG(1)G(2)G(3)-,-Cu(I)(H(2)O)(3)](+). The Cu(I) ion coordinates selectively to N7 of G(2) of 5'-G(1)G(2)G(3)-3' rather than N7 of G(1). The G(2)-selective coordination induces the G(1)-selective trap of a hole that is created by one-electron oxidation and migrates to GGG triplet. Therefore, the radical cation of G(1) is selectively created in both 5'-TG(1)G(2)G(3)-3' and 5'-CG(1)G(2)G(3)-3', giving the G(1)-selective damage of 5'-G(1)G(2)G(3)-3'.