The electrostatic calculation of molecular energies - II. Approximate wave functions and the electrostatic method

1954 ◽  
Vol 226 (1165) ◽  
pp. 179-192 ◽  
Keyword(s):  
Conventional Method ◽  
Hydrogen Molecule ◽  
Electronic States ◽  
Wave Functions ◽  
Chemical Bonds ◽  
Variational Procedure ◽  
Charge Densities ◽  
Electrostatic Method ◽  
Approximate Wave ◽  
Consistent Application

The electrostatic method of calculating molecular energies introduced in the first paper of this series is investigated more closely. It is shown that, for wave functions obtained by a consistent application of the Ritz variational procedure (floating functions), the electrostatic method is equivalent to the conventional method in terms of the Hamiltonian integral. Such floating functions are used to investigate various electronic states of the hydrogen molecule and the hydrogen molecular ion, and to explain certain anomalies in previous calculations. A method for estimating charge densities in localized chemical bonds is outlined.

The electrostatic calculation of molecular energies - I. Methods of calculating molecular energies

1954 ◽  
Vol 226 (1165) ◽  
pp. 170-178 ◽  
Keyword(s):  
Complex Systems ◽  
Conventional Method ◽  
Hydrogen Molecule ◽  
Electronic States ◽  
Wave Functions ◽  
Hydrogen Molecular Ion ◽  
New Methods ◽  
Electrostatic Method ◽  
Conventional Calculation ◽  
Approximate Wave

The calculation of molecular energies from assumed approximate wave functions is discussed. It is shown that the conventional method, based on the Hamiltonian integral, is but one of several possible approximations, and that two other methods, the virial method and the electrostatic method, avoid the most serious difficulties encountered in a conventional calculation. The mathematical simplicity of the new methods makes them especially suitable for non-empirical calculations on complex systems. The electrostatic method is exemplified by detailed calculations on various electronic states of the hydrogen molecule and the hydrogen molecular ion.

APPROXIMATE MOLECULAR ORBITALS: IV. THE 3dδg AND 4fδu STATES OF H2+

1967 ◽  
Vol 45 (8) ◽  
pp. 2533-2542 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran ◽  
Sheila D. McPhee
Keyword(s):  
Perturbation Theory ◽  
Variational Methods ◽  
Excellent Agreement ◽  
Hydrogen Molecule ◽  
Wave Functions ◽  
Simple Criterion ◽  
Wide Range ◽  
Approximate Wave ◽  
Exact Calculations ◽  
Schrödinger Perturbation

Properties of the lowest even and odd δ states of the hydrogen molecule–ion have been calculated using approximate wave functions. These were derived using a combination of Rayleigh–Schrödinger perturbation theory and variational methods, which have been applied previously to calculate the corresponding wave functions of the lowest σ and π states. Our total molecular energies are in excellent agreement with the recent exact calculations of Hunter and Pritchard (1967). A simple criterion is suggested for judging the accuracy of the approximate orbitals, which indicates that all the molecular properties calculated will be accurate over a wide range of internuclear separations.

Stationary properties of approximate wave functions

1969 ◽  
Vol 47 (21) ◽  
pp. 2355-2361 ◽  
Author(s):  
A. R. Ruffa
Keyword(s):  
Wave Function ◽  
Total Energy ◽  
Helium Atom ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Mechanical Wave ◽  
Expectation Value ◽  
Charge Densities ◽  
Hartree Fock ◽  
Approximate Wave

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.

History and Cross-Disciplinary Perspective of Compositional Imaging and Mapping With Nexafs Microscopy

1998 ◽  
Vol 4 (S2) ◽  
pp. 154-155
Author(s):  
H. Ade
Keyword(s):  
Electronic Structures ◽  
Chemical Shifts ◽  
Electronic States ◽  
Inorganic Materials ◽  
Chemical Bonds ◽  
Fundamental Physics ◽  
X Ray ◽  
Disciplinary Perspective ◽  
Core Electrons ◽  
X Ray Absorption

In Near Edge X-ray Absorption Fine Structure (NEXAFS) microscopy, excitations of core electrons into unoccupied molecular orbitals or electronic states provide sensitivity to a wide variety of chemical functionalities in molecules and solids. This sensitivity complements infrared (IR) spectroscopy, although the NEXAFS spectra are not quite as specific and “rich” as IR spectra. The sensitivity of NEXAFS to distinguish chemical bonds and electronic structures covers a wide variety of samples: from metals to inorganics and organics. (There is a tendency in the community to use the term NEXAFS for soft x-ray spectroscopy of organic materials, while for inorganic materials or at higher energies X-ray Absorption Near Edge Spectroscopy (XANES) is utilized, even though the fundamental physics is the same.) The sensitivity of NEXAFS is particularly high to distinguish saturated from unsaturated bonds. NEXAFS can also detect conjugation in a molecule, as well as chemical shifts due to heteroatoms.

The electrostatic calculation of molecular energies - IV. Optimum paired-electron orbitals and the electrostatic method

1956 ◽  
Vol 235 (1201) ◽  
pp. 224-234 ◽  
Keyword(s):  
Simple Procedure ◽  
Wave Functions ◽  
Optimum Form ◽  
Method Of Calculation ◽  
Hartree Fock ◽  
Electron Orbital ◽  
Electrostatic Method ◽  
Valence States ◽  
Paired Electron ◽  
Electrostatic Interpretation

Equations which determine the optimum form of paired-electron orbitals are derived. It is shown that for large nuclear separations these equations become the Hartree-Fock equa­tions for appropriate valence states of the separated atoms. An electrostatic interpretation of chemical bonding is developed using optimum paired-electron orbital functions. For these wave functions this simple procedure yields results identical with those obtained by the conventional method of calculation based on the Hamiltonian integral. Numerical computations by the electrostatic method are also discussed.

From Atomic Orbitals to Molecular Orbitals and Chemical Bonds

2020 ◽  
pp. 150-187
Author(s):  
Jochen Autschbach
Keyword(s):  
Hydrogen Molecule ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Basis Set ◽  
Orbital Method ◽  
Chemical Bonds ◽  
Atomic Orbitals ◽  
Hartree Fock ◽  
Generalized Matrix

It is shown how an aufbau principle for atoms arises from the Hartree-Fock (HF) treatment with increasing numbers of electrons. The Slater screening rules are introduced. The HF equations for general molecules are not separable in the spatial variables. This requires another approximation, such as the linear combination of atomic orbitals (LCAO) molecular orbital method. The orbitals of molecules are represented in a basis set of known functions, for example atomic orbital (AO)-like functions or plane waves. The HF equation then becomes a generalized matrix pseudo-eigenvalue problem. Solutions are obtained for the hydrogen molecule ion and H2 with a minimal AO basis. The Slater rule for 1s shells is rationalized via the optimal exponent in a minimal 1s basis. The nature of the chemical bond, and specifically the role of the kinetic energy in covalent bonding, are discussed in details with the example of the hydrogen molecule ion.

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