Hellmann–Feynman theorem near the threshold

The features of the spectrum structure are considered for situations where some parameter λ of the N‐electron Hamiltonian reaches the threshold value η under which the discrete energy level falls into the continuous spectrum. The electron density properties are also studied. It is proved that for a sequence of the wave functions converging in energy to the lower bound of the continuous spectrum as λ approaches η the corresponding sequence of the electron densities converges to the density of the (N ? 1)‐electron ground state. The results generalize the Hellmann–Feynman theorem for the cases where only the one‐side energy derivatives exist or there is no limiting wave function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Generalized Hellmann–Feynman theorem in the Xα method

In the Xα method a generalized Hellmann–Feynman theorem has been derived, and its most important applications are presented. A formula for electronegativity using the definition of electronegativity given by Iczkowski and Margrave and the magnetic nuclear shielding factor and its relation to the electronegativity have been obtained. The virial theorem of the Xα method has also been derived with the aid of the Hellmann–Feynman theorem.     

On the theoretical foundation of Walsh's rules of molecular geometry in terms of the Hellmann–Feynman theorem

A new interpretation of the ordinate in a Walsh diagram for a polyatomic molecule is suggested in terms of the Hellmann–Feynman theorem. This makes use of the fact that in a single-configurational MO wave function the total one-electron density is the sum of individual densities in the occupied orbitals. Walsh-type diagrams have been constructed for three different molecules, water, ammonia and hydrogen peroxide. In H₂O and NH₃ calculation of the force, and thus of the energy, in terms of the valence angle, is made on the assumption that the central (heavy) atom is kept fixed while each of the lighter atoms moves in a plane containing the principal symmetry axis and the relevant bond, in a totally symmetric fashion; for H₂O₂ the two oxygen atoms are kept fixed. The angular correlation diagrams obtained reproduce the general features of those obtained by plotting Hartree–Fock MO energies as functions of the valence angles. The conclusion emerges that the force formulation provides a satisfactory pictorial basis for understanding molecular geometry in terms of the balance between the electron–nucleus attractive forces resulting from the charge densities in the occupied MO'S , and the nuclear repulsive forces. However, in the absence of highly accurate charge distributions such an approach is unsuitable for the quantitative prediction of molecular quantities such as valence angles, force constants or energy barriers.     

Extension of the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages

In the present article, we extended the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages, that is, to the thermal states. We obtained some new formulas which make possible expressing the thermodynamical observables of the system as functions of the moments of coordinates, as we see in a short example relating to the pseudoharmonical oscillator. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 159–165, 1998     

Unlinked cluster effects in limited CI calculations of molecular properties

The unlinked cluster effects in limited CI calculations of dipole moments and polarizabilities are estimated by using the approximate corrections due to Davidson and Siegbahn. The results obtained for FH, H₂O, NH₃, and CH₄ indicate that the limited CI results for molecular electric properties need to be corrected for the erratic treatment of unlinked clusters.     

Obtaining self–consistent wave functions which satisfy the virial theorem

The virial theorem has played an important role in applying quantum mechanics to chemical problems. It has served as one criterion of a satisfactory wave function and its consequences on chemical bonding, molecular structure, and substituent effects have been analyzed extensively. A common method of gaining compliance with the virial theorem is to introduce a “scale” factor which adjusts all distances by a factor η. Optimizing the scale factor through the variational principle produces a wave function satisfying the virial theorem. In the present paper it is shown that when this “scaling” procedure is applied to self-consistent wave functions, the virial theorem can be satisfied, but self-consistency is lost. Scaling generally has a small effect on the total energy, but the effects on the energy components (T, V_ne, V_ee, V_nn) can be two to three orders of magnitude larger and in the range of tens to hundreds of kcal. Consequently, for applications where the energy components are useful, it is highly desirable to obtain wave functions which satisfy the virial theorem and are self-consistent. In the present paper, a simple, inexpensive extrapolation technique is reported which requires one integral evaluation and two SCF cycles to achieve convergence. Applications to atoms and small molecules are reported.     

New derivation of the Waller–Hartree–Fock spatial wave function

A new derivation is given for the Waller–Hartree–Fock double-determinantal spatial wave function. One starts from the single-determinant wave function in which a orbitals are doubly occupied, and decomposes it into a sum of products of spatial and spin functions. The spatial product of the first genealogical spin eigenfunction is a double-determinantal function. The derivation is based on the simple form of U_1?(P) when the representation matrix is obtained from the genealogical spin eigenfunction.     

Molecular integrals in the generalized hylleraas–CI method

In the generalized Hylleraas–CI method, the original correlation factor r^v_ij is multiplied by a Gaussian geminal. Using the approach of generating functions, the general formulas of molecular integrals in this method are derived over Cartesian Gaussian orbitals. From differentiations of the generating functions, the expanding length in the incomplete Gamma functions is reduced, and some cancellations presented in other approaches are avoided. Preliminary calculations for H₂ and H₂—H₂ systems are carried out over STO -3G basis. The results are encouraging.     

Cluster expansion of the wave function. Electron correlations in singlet and triplet excited states,ionized states,and electron attached states by SAC and SAC–CI theories

The symmetry-adapted-cluster (SAC ) and SAC –CI theories reported previously have been applied to the study of electron correlations in ground state, singlet and triplet excited states, ionized state, and electron attached state. Formulas for calculations of one-electron properties and transition properties from the SAC and SAC –CI wave functions are given. Calculations were carried out for the ground and Rydberg excited states of water and its positive and negative ions, with the use of the simpler computational scheme than the previous one. The results compare well with experiments.     

Investigation of excited‐state properties of fluorene–thiophene oligomers by the SAC‐CI theoretical approach

Excited states of fluorene‐ethylenedioxythiophene (FEDOT) and fluorene‐S,S‐dioxide‐thiophene (FTSO2) monomers and dimers were studied by the symmetry‐adapted cluster (SAC)‐configuration interaction (CI) method. The absorption and emission peaks observed in the experimental spectra were theoretically assigned. The first three excited states of the optimized conformers, and the conformers of several torsional angles, were computed by SAC‐CI/D95(d). Accurate absorption spectra were simulated by taking the thermal average for the conformers of torsional angles from 0° to 90°. The conformers of torsional angles 0°, 15°, and 30° mainly contributed to the absorption spectra. The full width at half‐maximum of the FEDOT absorption band is 0.60 eV (4839 cm^?1), which agrees very well with the experimental value of 0.61 eV (4900 cm^?1). The maximum absorption wavelength is located at 303 nm, which is close to those of the experimental band (327 nm). The calculated absorption spectrum of FTSO2 showed two bands in the range of 225–450 nm. This agrees very well with the available experimental spectrum of a polymer of FTSO2, where two bands are detected. The excited‐state geometries were investigated by CIS/6‐31G(d). These showed a quinoid‐type structure which exhibited a shortening of the inter‐ring distance (0.06 Å for FEDOT and 0.04 Å for FTSO2). The calculated emission energy of FEDOT is 3.43 eV, which agrees very well with the available experimental data (3.46 eV). The fwhm_E is about 0.49 eV (3952 cm^?1), while the experimental fwhm is 0.43 eV (3500 cm^?1). For FTSO2, two bands were also found in the emission spectrum. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Numerical studies for a theoretical analysis of semiempirical LCAO–CI methods

A numerical investigation of Del Re and Parr's formulas [1] for the treatment of π systems has been preformed in the case of five-membered rings, using two different expressions for the core Hamiltonian and different values for the effective charges. The results obtained are discussed by analysing the three stages of the calculation: (a) a non-iterative LCAO –MO calculation; (b) the same calculation with corrections for exchange and repulsion terms arising from fluctuations of the orbital populations; (c) configuration interaction. The calculations are interesting also because they do not involve the zero differential overlap approximation; a calculation without inclusion of overlap hse been carried out for pyrrole and the results have been compared with those including S . The main conclusions hold also for σ electrons, and can serve to assess better the validity of simple σ calculations.     

The electronic structure and one-electron properties of BH computed from SCF and CI wave functions

The SCF and CI wave functions for BH, obtained in calculations described in detail elsewhere [2], are compared through their electron distributions and electron moments.Taken in part from a Ph.D. thesis submitted to the University of Toronto in 1971.     

Modifications of virtual orbitals in the limited CI calculations for electron-rich molecules

The efficiency of modified virtual orbitals (MVO) of ionic type and of approximate orthogonalized natural orbitals (ONO) in the CI-SD calculations was studied for O₃ and SO₂ molecules and compared with the commonly used canonical virtual orbitals (CVOs). The systems studied represent a class of electron-rich molecules, in which the number of valence electron pairs exceeds substantially the number of formal chemical bonds. We found that the modified orbitals of the types studied appear to be less effective for these systems than in the similar calculations for the AH_n type molecules. Physical reasons for this difference were discussed. The evolution of spatial properties of virtual orbitals within the modification process was analyzed. © 1996 John Wiley & Sons, Inc.     

The convolution theorem and the Franck–Condon integral

The convolution theorem is used to evaluate the Franck–Condon integral. It is shown that this integral becomes the matrix element between two “squeezed” states. This enables one to evaluate the integral by using boson operators. In addition, a general method is developed to obtain integrals involving Hermite polynomials with a displaced argument. In particular, the two‐center matrix element _g〈m|f(x_e)|n〉_e, is obtained, where f(x_e)=exp(Dx+Fx_e). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 11–15, 1999     

Effects of wave function modifications on calculated carbon–hydrogen bond lengths

The two-level factorial design (FD) and principal component analysis (PCA) chemometric techniques were used to investigate the carbon–hydrogen bond lengths dependence on the basis set size and quantum chemistry method, for H–C≡CH, H–C≡CF, H–C≡CCH₃, H–C≡CCN, H–C≡CCl and H–C≡CCCH molecular systems. The calculations were performed by using Hartree-Fock (HF), Møller-Plesset 2 (MP2) and Density Functional Theory (DFT) with B3LYP exchange-correlation functional methods. The effects concerning basis set size include the number of valence and polarization functions as well as the cooperative effect between them, at all computational levels. The increase in the number of valence functions decreases the calculated C–H bond lengths by approximately 0.0022 Å, while the inclusion of polarization functions at HF and B3LYP levels increases the C–H bond length, in contrast to the behavior obtained at MP2 level. The effect of the inclusion of diffuse functions is non-significant, at all three computational levels. Moreover, the valence–polarization interaction effects are not significant, except at the MP2 calculational level, in which such effects lead to an increase in the calculated C–H bond lengths. When the computational level changes from HF→B3LYP and B3LYP→MP2 the calculated C–H bond length values increase (on average) by +0.0100 and +0.0027 Å, respectively. Algebraic models (one for each level of calculation) successfully employed to reproduce the calculated values for H–C≡N bond length, a system not included in the training set. The HF/6-31G(d,p) and HF/6-31++G(d,p) results yield the lowest standard errors (0.0015 and 0.0014 Å, respectively) and correspond to the calculated points in closest proximity to the experimental one.     

SCF CI MO calculations on the base–base interactions in dinucleoside monophosphates

In order to elucidate further details of RNA conformations we have studied base stacking in dinucleoside monophosphates (DNP's) using UV difference spectra and the hypochromic effect. SCF CI MO calculations according to PPP and MIM approximations were carried out for six pyrimidine-containing DNP's in each of several stacking geometries. The calculated and plotted difference spectra were fitted to the experimental spectra. Different DNP's showed distinct geometries in the range of the helix angle of 35°. From our results we conclude that there is a microstructure in the helix. This would imply an additional content of information in this macromolecule, a higher precision in nucleic acid interactions, and make possible the prediction of the conformation of polynucleotides.     

Monte Carlo simulations of solid nitrogen in the isothermal–isobaric ensemble with an ab initioSCF–CI potential

Monte Carlo simulations in the isothermal–isobaric ensemble for the α phase of solid N₂ have been carried out with two different pair potentials obtained from ab initio quantum chemical calculations. Comparison is made with data obtained from empirical potentials as well as with experimental data, and in general, the agreement with experiment is found to be good. It is also found that the differences between experiment and theory can largely be attributed to correlation effects, not considered in the quantum mechanical calculations of the pair potential.     

MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem

A novel approach is developed for optimizing molecular orbitals within the context of a multiconfiguration self-consistent-field problem. The MCSCF wave function is determined through a sequence of eigenvalue problems in the multiconfiguration space and the single-excitation space. They are used to iteratively improve the natural orbitals, which in turn are related, by successively improved transformations, to the MCSCF orbitals. The mathematical problems arising out of this general concept are solved and the computational implementation is discussed. In many applications the method has proven itself as a powerful approach in forcing rapid convergence. Adaptation to spin and spatial symmetry is maintained throughout and the procedure is applicable to excited states as well as to ground states.