Coupled Hartree-Fock Perturbation Research Articles

Extension of the Genkin and Mednis treatment for dynamic polarizabilities and hyperpolarizabilities of infinite periodic systems. I. Coupled perturbed Hartree–Fock theory

The uncoupled theory of Genkin and Mednis [Sov. Phys. JETP 27, 609 (1968)] for the nonlinear optical properties of infinite periodic systems has been extended to yield a fully analytical coupled perturbed Hartree–Fock treatment. Similarities and differences from other approaches to the same problem are analyzed. Future addition of electron correlation and vibrational contributions is discussed.

Accurate predictions for molecular magnetisabilities and rotational g tensors using a simple and efficient DFT approach

In a recent paper, we applied a simple but highly efficient density functional theory (DFT) method which appeared to correct for the deficiencies in the virtual Kohn–Sham (KS) orbitals and eigenvalues of the prediction of nuclear shielding parameters. We now report the performance of this new DFT approach to the prediction of molecular magnetisabilities and rotational g tensors. For comparison, large multiconfigurational self-consistent field (MCSCF) methods are available as very accurate benchmark values.The results are very encouraging. The new approach produces a significant improvement over the coupled-perturbed Hartree–Fock (CPHF) SCF method and the conventional DFT functionals for the prediction of both isotropic and anisotropic magnetisabilities. For the latter, our results approach the accuracy of the reference MCSCF values. For the rotational g tensors, DFT methods are found to be greatly superior to the CPHF SCF values. Comparison with the reference MCSCF values indicates that the new functionals provide superior results compared to conventional DFT functionals.

Adiabatic correction to the energy of molecular systems: the CPHF equivalent of the Born–Handy formula

The Born–Handy formula, recently shown by Kutzelnigg to be a rigorous expression for the calculation of the adiabatic correction, has been, on the level of a ground state SCF wavefunction, reformulated and linked to the coefficients of the standard coupled perturbed Hartree–Fock (CPHF) method. The contribution of the electron correlation via second-order perturbation theory is also presented. The solution of the corresponding secular equation of the nuclear motion enables the calculation of the adiabatic correction over the particular normal modes. The method offers the possibility of extending high-precision calculations of the adiabatic correction to more complex systems. Test calculations have been performed for H 2, HD and D 2 and the results are in satisfactory agreement with the exact figures.

Theoretical Investigation of the Second and Third Order Nonlinear Optical Properties of Some Fused Heterocyclic Aromatic Compounds

ABSTRACTWe report herein the results of coupled perturbed Hartree-Fock (CPHF) ab initio extended basis set calculations on the geometric structures, dipole moments, static first-order (α), second-order (β), and third-order polarizabilities (λ) of a series of fused heterocyclic aromatic compounds based on quinoline. The effects of the presence/absence of the heteroatom as well as the introduction of other substituents at various positions in the ring system on these molecular properties are described. The effect of the presence of N-oxide is also examined. Suggestions for the design of heterocyclic systems with enhanced polarizabilities are made.

Evaluation of analytic molecular orbital derivatives and gradients using the effective valence shell Hamiltonian method

Expressions for the analytic energy gradients and the nonadiabatic derivative couplings are derived for the effective valence shell Hamiltonian theory (a variant of degenerate/quasidegenerate many-body perturbation theory) using the diagonal and off-diagonal Hellmann–Feynman formulas and a generalized set of coupled perturbed Hartree–Fock equations to evaluate the derivatives of the molecular orbitals. The method is designed for efficiently treating the energy derivatives and nonadiabatic couplings for several states simultaneously. The generalized coupled perturbed Hartree–Fock equations arise because the reference space orbitals are optimized for simultaneously describing the ground and excited states, a feature lost with the traditional partitioning where the virtual orbitals provide a poor choice for representing the low lying states. A simple correspondence emerges between the new generalized coupled perturbed Hartree–Fock and the traditional coupled-perturbed Hartree–Fock methods enabling the use of the former with straightforward modifications. The derivatives of the second and higher order portions of the effective Hamiltonian are readily obtained using a diagrammatic representation that will be described elsewhere.

Analytical evaluation of energy derivatives in extended systems. I. Formalism

A method is developed to analytically evaluate energy derivatives for extended systems. Linear dependence among basis functions, which almost always occurs in extended systems and brings instability to the coupled-perturbed equations, is automatically eliminated in this method. The remaining independent basis functions are transformed into semiorthogonal orbitals. The derivatives of the orbitals and the overlap matrix over them are obtained via a set of coupled-perturbed equations, similar to those of the coupled-perturbed Hartree-Fock (CPHF) equations which are used to calculate the derivatives of the Hartree-Fock (HF) orbitals and the orbital energies. By introducing symmetrized coordinates, these coupled-perturbed equations can be easily solved. Explicit expressions for calculating gradients and Hessians of the HF energy for extended systems are given. With this method, we can calculate energy derivatives with respect to displacements of the nuclei, including those which break the translational symmetry. Therefore, the method not only provides an efficient and accurate approach to calculate energy derivatives of any order, but also enables us to determine the force constants for individual nuclei, the interatomic force constants, and phonon dispersion curves in the whole Brillouin zone. With this method, the computational cost to calculate phonon spectrum with k≠0 in the Brillouin zone is the same as that needed for the spectrum at k=0.

Analytical second derivatives in ab initio Hartree–Fock crystal orbital theory of polymers

In the framework of ab initio Hartree–Fock crystal orbital theory of polymers, the formulas for the analytical second derivatives of energy with respect to in-phase ( k=0) nuclear coordinates are derived. The coupled perturbed Hartree–Fock (CPHF) equation is iteratively solved by using the direct (recomputation of two-electron integrals) atomic-orbital-based algorithm. Frequencies of the Brillouin zone center ( k=0) vibrations of all- trans polyethylene are calculated by using the STO-3G, 3-21G and 6-31G* basis sets. The dependence of the frequencies on the number of neighbors included in the lattice summations, on the number of momentum sampling points in the first Brillouin zone, and on the convergence criterion for the CPHF solutions is examined. In our implementation, the use of analytical second derivatives is more efficient than the use of the finite differences of analytical first derivatives.

Calculation of the NMR shielding of O in MgO at high temperature

17O NMR studies indicate that the O in crystalline MgO becomes less shielded magnetically as the temperature is increased. Over a range of about 1300 °C the shielding decreases by about 10 ppm. Increasing the temperature in MgO also leads to a small increase in the average Mg-O bond length and a large increase in the isotropic temperature factor, corresponding to a large increase in the root mean square deviation of the Mg-O distance from its equilibrium value. Using the gauge-including-atomic-orbital (GIAO) version of coupled Hartree-Fock perturbation theory, implemented at the self-consistent-field (SCF) level using a polarized split valence basis set, we have calculated the NMR shielding at the central O of a cluster OMg6(OH)12 −2, Oh symmetry, for several values of the Mg-O distance within a range of 30% about our calculated equilibrium value. We have also considered some distorted geometries for this cluster. We find that the O NMR shielding decreases nonlinearly as the central Mg-O distance is decreased. For the shortest Mg-O distance studied (1.391 A) in the symmetric geometry the calculated O NMR shielding was 229.4 ppm, while at the equilibrium Mg-O distance calculated for the cluster (1.987 A) the calculated shielding was 317.8 ppm. If the average 17O shielding is estimated by weighting the calculated NMR shieldings at various Mg-O distances by the probability of that distance, as obtained from the experimental isotropic temperature factors, at least a part of the observed deshielding at high temperatures can be reproduced. Calculations on a C4v symmetry OMg6(OH)12 −2 cluster, with the central O displaced from the centre, show even stronger deshielding. Therefore, consideration of such asymmetric motions within the central OMg6 unit could further reduce the calculated 17O shielding.

Ab initio prediction of 15N‐NMR chemical shift in α‐boron nitride based on an analysis of connectivities

Hydrogen-saturated cut-outs of hexagonal boron nitride have been used to model the solid state. Model compounds have been geometry optimized by means of density functional theory, whereas chemical shift calculations have been carried out at the coupled-perturbed Hartree–Fock level of theory employing gauge-including atomic orbital (GIAO) basis sets. The reliability of results has been tested against experimental values for chemical shifts in stable molecules with similar structural elements. With increasing cluster size, viz. a vanishing influence of the saturating hydrogens on the innermost nitrogen atoms, we find a convergence of 15N chemical shifts. A classification scheme for the chemical environment of a nitrogen atom has been set up according to its bonding graph including the second coordination sphere. For a given connectivity, chemical shifts vary within a few parts per million, thus enabling us to predict a 15N-NMR chemical shift of −285 ± 5 ppm for solid α-boron nitride. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 716–725, 1998

Ab initio prediction of15N-NMR chemical shift in ?-boron nitride based on an analysis of connectivities

Hydrogen-saturated cut-outs of hexagonal boron nitride have been used to model the solid state. Model compounds have been geometry optimized by means of density functional theory, whereas chemical shift calculations have been carried out at the coupled-perturbed Hartree–Fock level of theory employing gauge-including atomic orbital (GIAO) basis sets. The reliability of results has been tested against experimental values for chemical shifts in stable molecules with similar structural elements. With increasing cluster size, viz. a vanishing influence of the saturating hydrogens on the innermost nitrogen atoms, we find a convergence of 15N chemical shifts. A classification scheme for the chemical environment of a nitrogen atom has been set up according to its bonding graph including the second coordination sphere. For a given connectivity, chemical shifts vary within a few parts per million, thus enabling us to predict a 15N-NMR chemical shift of −285 ± 5 ppm for solid α-boron nitride. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 716–725, 1998

Analytical energy gradients for local second-order Mo/ller–Plesset perturbation theory

Based on the orbital invariant formulation of Mo/ller–Plesset (MP) perturbation theory, analytical energy gradients have been formulated and implemented for local second order MP (LMP2) calculations. The geometry-dependent truncation terms of the LMP2 energy have to be taken into account. This leads to a set of coupled-perturbed localization (CPL) equations which must be solved together with the coupled-perturbed Hartree–Fock (CPHF) equations. In analogy to the conventional non-local theory, the repeated solution of these equations for each degree of freedom can be avoided by using the z-vector method of Handy and Schaefer. Explicit equations are presented for the Pipek–Mezey localization. Test calculations on smaller organic molecules demonstrate that the local approximations introduce only minor changes of computed equilibrium structures.

Effects of main chain conformation on third-order non-linear optical susceptibilities of polysilanes Theoretical study on their second hyperpolarizabilities and bulk susceptibilities

Static third-order non-linear optical susceptibilities χ(3) of polysilanes were estimated at molecular and crystalline levels in order to examine effects of main chain conformation on their χ(3) values. A combination of an abinitio coupled perturbed Hartree–Fock (CPHF) method and the oriented-gas approximation, which was previously developed by the author (T. Hamada, J. Chem. Soc., FaradayTrans., 1996, 92, 3165), was employed for the estimation. Molecular calculation results showed cis and gauche Si–Si bonds reduce γ of polysilanes, and trans-planar polysilanes have larger second hyperpolarizabilities γ (molecular χ(3)) than trans–cis, alternating trans–gauche, and 7/3 helix polysilanes. On the other hand, crystal calculation results indicated χ(3) of polysilane crystals are influenced not only by the main chain conformation of the polysilanes, but also by the molecular weight of their repeating unit (MSi). These findings were used to propose a molecular design method for polysilane χ(3). The method showed trans-planar poly(dimethylsilane) (PDMS) and poly(diethylsilane) (PDES) are attractive materials for experimental studies, owing to their enhanced χ(3) values.

On the calculation of parity-violating energies in hydrogen peroxide and hydrogen disulphide molecules within the random-phase approximation

It is shown via extended numerical tests that the ab initio calculation of a parity-violating energy term, at the random-phase approximation (or coupled Hartree-Fock perturbation theory) level of accuracy, gives results which are more than one order of magnitude larger than those usually obtained by means of less accurate methods employed so far. These findings make more plausible the hypothesis of electroweak selection of natural enantiomers.

Calculation of B and O NMR parameters in molecular models for B 2O 3 and alkali borate glasses

To help in the interpretation of the 11B and 17O magic-angle-spinning (MAS) and double-rotation nuclear magnetic resonance (DOR NMR) spectra of B 2O 3 and related glasses we have calculated NMR parameters from ab initio self-consistent-field MO theory for molecules modeling the local B and O environments of the glasses. The present work employs larger model molecules, larger basis sets, and a better quantum mechanical method (the gauge-including atomic orbital version of coupled Hartree-Fock perturbation theory) than in my previous studies. The calculated 11B shieldings confirm the deshielding of the B atoms within boroxol rings, B 3O 3(-O) 3, compared to corner sharing BO 3 units. The shielding of B atoms within the boroxol rings is almost unchanged when two or even three boroxol rings are linked together, but the shielding of B atoms in the corner-sharing BO 3 units is quite sensitive to the size of the cluster. Calculated O NMR shieldings and quadrupole coupling constants indicate that O in boroxol rings is deshielded and has a smaller coupling constant than O bridging between triangular BO 3 units. Calculated 17O shieldings are smaller for atoms in the middle of a cluster than for chemically similar atoms on the periphery. O atoms bridging between two boroxol rings or between a boroxol ring and a BO 3 unit have intermediate values for both shielding and coupling constant. 17O shieldings show a considerable range and seem more sensitive to distant neighbors than do the 11B shieldings. Perturbations of the boroxol ring model by deprotonation, or by interaction with Na + or OH −, change the 11B shielding by only 1 ppm, while the 17O shieldings and the frequencies of the Raman active symmetric ring O breathing modes show substantial changes. Thus, boroxol ring-like 11B NMR signatures may remain while boroxol-ring Raman spectral signatures disappear.

Theoretical estimates of the rotational g-factor, magnetizability and electric dipole moment of GaH

The rotational g-factor, magnetizability and electric dipole moment of GaH were calculated using several perturbative and multiconfigurational polarization propagator approximations. The coupled perturbed Hartree-Fock (CHF) result for the rotational g-factor ( g J = −3.245 for 69Ga 1H at the experimental equilibrium geometry) is in good agreement with a recent estimate of the rotational g-factor from non-adiabatic rotational effects observed in field-free vibration-rotation spectra of GaH. However, no agreement is found between the experimental estimate and the results of calculations including electron correlation ( g J = −3.380 at the level of the second order polarization propagator approximation with coupled-cluster singles and doubles amplitudes — SOPPA(CCSD) — and g J = −2.946 using a multiconfigurational polarization propagator approach (MCPPA)). Averaged values of g J for the v = 0, J = 1 and v = 1, J = 1 ro-vibrational states are reported for all isotopomers, which are less negative than the results for the equilibrium geometry. GaH is predicted to be diamagnetic. The results for the electric dipole moment are in disagreementwith the estimate from non-adiabatic rotational effects. Ro-vibrational averaging increases the absolute value of the the dipole moment.

Ab initio molecular orbital calculations of the static polarizabilities of xanthone analogues

The static polarizabilities, α, of various xanthone analogues ( 1–19) were estimated by ab initio molecular orbital calculations using the coupled perturbed Hartree-Fock (CPHF) method. The influence of basis sets on the calculated values was examined in detail and the reliability of the ECP approach was confirmed. A good linear relationship was found between the Δα values calculated using the 3–21G basis set and those using larger basis sets. The introduction of substituents generally increases 〈α〉, whereas Δα is strongly affected by the nature of the substituents and by the molecular geometries. According to Prasad's equation, γ orient was calculated from Δα and compared with experimental values.

Quadratic Nonlinear Optical Susceptibilities of a New SHG Material, 3-Aminoxanthone: A Theoretical Study on Its Molecular and Crystal Susceptibilities

Nonlinear optical susceptibilities of a new SHG (second harmonic generation) material, 3-aminoxanthone, were investigated at both molecular and crystalline levels, by using the ab initio CPHF (coupled perturbed Hartree−Fock) method and the oriented-gas approximation. Molecular first hyperpolarizabilities β of 3-aminoxanthone were analyzed in order to investigate the source of β, by using an intramolecular oriented-gas approach which assumes the molecular β can be obtained by making a tensor sum of the β of molecular fragments. The analysis revealed the source of β is localized on one aromatic ring, and effective π-conjugation for β is almost completely disconnected between the aromatic rings. The results of the oriented-gas calculations showed the crystal has a phase matching condition of SHG, and its is 10.6 times that of the of urea crystal; these results are in agreement with the results of the SHG powder efficiency measurements. Experimental refractive indices were needed for the local field correctio...

The sum of hyperpolarizabilities approach to third-order non-linear optical susceptibilities of π-conjugated polymer crystals

Non-resonant third-order non-linear optical susceptibilities χ(3)zzzz are estimated for infinite conjugation lengths of polymers, such as polyacetylene, polydiacetylene and poly-yne in their crystal states, by using the calculated static second hyperpolarizability γ and the sum of hyperpolarizabilities approach. The static second hyperpolarizability of the oligomers is calculated by using an ab initio coupled perturbed Hartree–Fock (CPHF) method. The CPHF calculation for the polyacetylene oligomer reproduces the experimental values of the longitudinal component γzzzz of γ observed at non-resonant wavelengths. The static second hyperpolarizability γ of the infinite polymers is extrapolated from those of their oligomers considering the saturation behaviour of γ in a longer chain region. This combination of the CPHF calculation and the sum of hyperpolarizability approach is successfully applied to the evaluation of χ(3)zzzz of bulk polyacetylene. However, the above method cannot correctly reproduce χ(3)zzzz of polydiacetylene crystal. The results are discussed considering the difference between polyacetylene and polydiacetylene; i.e., excitonic or non-excitonic nature of their excited states.

Excited-state gradients viaCPHF equations

A detailed derivation of the equations for the energy gradients for excited-state wave functions is presented. These excited states are generated by singles-only CI (CIS) from the Hartree–Fock ground-state wave function. The gradients are obtained from direct differentiation of the CIS energy expression. Since this expression contains explicit terms involving the derivative of the Hartree–Fock orbital coefficients, coupled perturbed Hartree–Fock (CPHF) equations are used to eliminate the need to evaluate these terms. The final equations are then transformed to the atomic orbital basis to avoid costly AO-to-MO transformations. © 1995 John Wiley & Sons, Inc.

Ab initio studies of the polarizabilities of retinal analogs

Axial components of the polarizability, hyperpolarizability, and second order hyperpolarizability of retinal model molecules are calculated using the ab initio coupled-perturbed Hartree–Fock theory. Conformational rearrangements are shown to affect some components in a systematic fashion. We also find that the general trend in the hyperpolarizability behavior as a function of protonation and charge transfer is in agreement with available experimental data. The second hyperpolarizability of the protonated Schiff base is found to be highly dependent on the adopted geometry optimization. We explore a possible explanation for this as well as present a brief discussion with regard to future experimental and theoretical developments towards the construction of molecularly based nonlinear optical devices.