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.     

The Hellmann–Feynman theorem for statistical averages

We discuss the Hellmann–Feynman theorem for degenerate states and its application to the calculation of the derivatives of statistical averages with respect to external parameters.     

Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential

We solve the D‐dimensional Schrödinger equation under the Hua potential by using a Pekeris‐type approximation and the supersymmetry quantum mechanics. The reliability of the spectrum is checked via a comparison with the finite difference method. This interaction resembles Eckart, Morse, and Manning–Rosen potentials. Some useful quantities are reported via the Hellmann–Feynman Theorem. © 2012 Wiley Periodicals, Inc.     

Constraints and the Hellmann-Feynman theorem

It is pointed out that Hurley's condition applies even in the presence of Lagrange multipliers. Also a useful formula for E is derived which allows for the possibility that both the trial functions and the constraints may depend on .     

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.     

First-order properties and the Hellmann–Feynman theorem in the case of a limited CI wave function

Two distinct approaches to the calculation of first-order properties with a limited CI wave function are discussed. One is based on the Hellmann–Feynman theorem and the other on the direct evaluation of the total energy derivative at zero perturbation. Corrections to the Hellmann–Feynman expectation value are given for the CI wave function consisting of a single determinant reference state and all single and double replacements of this. These corrections are the extended Brillouin matrix elements and involve interactions between the zeroth-order wave function and triply substituted configurations. The usefulness of these matrix elements for the generation of MC SCF orbitals and for the calculation of cluster corrections to the wave function is briefly discussed. The formulas for the Brillouin matrix elements expressed in terms of one- and two-electron integrals have been automatically generated using the syntax of the algebraic program SCHOONSCHIP.     

The computation of floating functions and their use in force constant calculations

It is shown how standard computer programs for calculating the equilibrium geometry of a molecule may be adapted to yield floating functions. These functions satisfy the Hellmann–Feynman theorem and so lead to simple electrostatic interpretations of intramolecular forces and vibrations. The theory is illustrated by detailed calculations for the water molecule.     

Many-electron problems III scaling of the density distributions and the potential fields

The Hellmann–Feynman theorem has been used to prove some energy relations. In this paper it is shown that in a scaled co-ordinate system both the charge density and the potential field of neutral atoms may be scaled so that the corresponding scaled quantities are independent of the atomic number. In the proof the momentum theorem of real functions is used.     

An improved Hellmann-type pseudopotential for atoms and molecules

The concept of pseudopotentials offers much attractiveness for the quantum mechanical evaluation of the physical properties of atoms and molecules. The ideas of Hellmann, in which the repulsive and fermion character of inner electrons can be mimicked by an experimentally fitted, exponentially damped potential term, are especially attractive. Unfortunately, it is found that such a simple expression can only be used in a very limited number of cases, such as for the alkali metals, and even then fails for the simple case of lithium. The present study shows that the Hellmann idea can readily be extended by including a second “shielded potential” term evaluated from tabulated previous Hartree-Fock calculations. The new expression for the model pseudopotential is both simple and effective. With it, the inner potential of any of the alkali metal atoms, including lithium, can be represented so that calculation of the molecular properties of the metal dimers can be accomplished. Calculations for Li₂, Na₂, and K₂ show the binding energies and equilibrium interatomic distances to be quite well given, in agreement with both chemical experience and spectroscopic evidence.     

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     

Momentum density interpretation of the H(1s)–H+ long-range force

The recently proposed method of using momentum densities for interatomic interactions is applied to the long-range force between the ground-state hydrogen atom and the proton, and the results are compared with those from using the position density based on the electrostatic Hellmann–Feynaman theorem. A new physical interpretation of the long-range force is obtained, which is complementary to that in position space. It is found that some perturbative changes in the position density do not accompany changes in the momentum density.     

Four-dimensional density and energy density functional

The relations based on an external one-electron operator V( r ) are examined from two view-points, i.e., from the Hohenberg–Kohn approach and the four-dimensional density concept introduced by Wilson and Frost, and extensively studied by Parr and Politzer. The object being to obtain, with the help of the Hellmann–Feynman theorem, new formulas for the energy of atoms and molecules, and to discuss the construction of the universal energy density functional on the basis of the four-dimensional density.     

A general procedure for obtaining wave functions obeying the virial theorem

The virial theorem for molecules is shown to have two different forms, one employing the energy gradient the other involving the Hellmann–Feynman force. While the former VT can be fulfilled by a uniform scaling of the basis set, the latter cannot be satisfied in certain basis sets, and can give unrealistic results in others. The scaling procedure is applied to molecules at nonstationary points on the potential energy surface and it is found that energy components can change substantially, especially at short bondlengths, while the change in total energy is small. The effects on molecular properties are also small.     

Hans‐Hellmann‐Symposium in Nowgorod

Zum sechsten Male trafen sich Mitte Mai die Vertreter der Theoretischen Chemie Rußlands anläßlich der V. A. Fock School on Quantum and Computational Chemistry 180 km südöstlich von St. Petersburg in Nowgorod, in der Staatlichen Universität Yaroslav der Weise. Die diesjährige Konferenz übertraf mit rund 100 Beiträgen deutlich die des Vorjahres. Beim erstmals abgehaltenen Hans‐Hellmann Symposium ging es um eine kritische Sichtung und Weiterentwicklung der Dichtefunktionaltheorie.     

Theoretical energies of representative carbon–carbon bonds

A bond-energy formula deduced by means of the Hellmann–Feynman theorem is applied to selected simple hydrocarbons. The required potentials at the nuclei are calculated with the help of large basis-set expansions including polarization functions. The carbon–carbon bond energy of ethane is evaluated at ～ 70 kcal mol^?1. The CC bond energies of ethane, ethylene, acetylene, benzene, and cyclopropane are approximately in a ratio of 1: 2.0: 3.0: 1.65 1.0. Limitations and possible improvements in future applications of this energy formula are discussed. © 1995 John Wiley & Sons, Inc.     

Topology of density difference and force analysis

The density difference (r) of a molecule A-B is defined as the difference of the density (r) of the molecule A-B and the density _A(r) + _B(r) put at the position of the atoms A and B. We investigate here the topological features of the density difference and define electron density flow (EDF) as representing the direction and the amount of the electron density flow in the course of the nuclear displacement processes. As such examples, we study H₂ molecule formation reaction and the interaction of two He atoms. By the topological analysis of (r), and by using the Hellmann-Feynman force and its partition into the AD, EC, and EGC forces, the characteristic behaviors of the (r) map are clarified. In particular, the electron cloud preceding and incomplete following are represented by using the concept of the EDF. The natures of the covalent bond are clarified based on the topological properties of the difference density (r) rather than that of the total density (r).On leave from the Department of Chemistry, Hebei Teachers' College, Shijiazhuang, Hebei, 050091, China     

Zur Konstitution des Produktes aus Benzyliden-bis-acetamid und Dihydroresorcin

Zusammenfassung Das aus Benzyliden-bis-acetamid und Dihydroresorcin erhaltene Reaktionsprodukt¹ ist nicht das 2-Benzyliden-cyclohexan-1,3-dion, sondern mit dem erstmals vonD. Vorländer undO. Strauss ⁷ hergestellten 9-Phenyl-1,2,3,4,5,6,7,8-octahydroxanthen-1,8-dion identisch.the author is geb. A dametz H. Hellmann, G. Aichinger undH.-P. Wiedemann, Ann. Chem.626, 35 (1959).     

Statistical thermodynamic generalization of the Hellmann—Feynman and quantum mechanical curvature theorems

The statistical thermodynamic analogs and generalization of the Hellmann—Feynman theorem and the quantum mechanical curvature theorem of Silverman and van Leuven are derived for hamiltonians of the form Σ_m=1^MP_mHP_m the P's forming a complete set of mutually orthogonal and parametrically-dependent projections which maximize the associated partition functions.     

Energy levels for perturbed Morse oscillators

The energy levels and perturbation expansions for the expectation values of arbitrary powers of position for a perturbed Morse oscillator are obtained by application of the hypervirial and Hellmann–Feynman theorems, solely in terms of the unperturbed energy. We obtain expressions for the first-order corrections for (1 ? e^?aq )^m for 4 ? m ? 8 and the expressions to second and third order for the quartic perturbation. A numerical application to the CO molecule is made.