Projection of the wave function of the unlimited Hartree-Fock method on the doublet state in the case of benzyl radical

1974 ◽  
Vol 8 (3) ◽  
pp. 242-249
Author(s):  
I. I. Ukrainskii ◽  
Yu. A. Kruglyak ◽  
H. Preuss ◽  
R. Yanoshek
Keyword(s):  
Wave Function ◽  
Doublet State ◽  
Hartree Fock ◽  
Benzyl Radical

scholarly journals Evaluation of the one electron expectation values for different wave function of Be atom

2007 ◽  
Vol 4 (3) ◽  
pp. 393-396
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Slater Determinant ◽  
Open Shell ◽  
Expectation Values ◽  
Electronic Density ◽  
Expectation Value ◽  
Hartree Fock ◽  
Shell System ◽  
Electronic Wave Function ◽  
The One

The aim of this work is to evaluate the one- electron expectation value from the radial electronic density function D(r1) for different wave function for the 2S state of Be atom . The wave function used were published in 1960,1974and 1993, respectavily. Using Hartree-Fock wave function as a Slater determinant has used the partitioning technique for the analysis open shell system of Be (1s22s2) state, the analyze Be atom for six-pairs electronic wave function , tow of these are for intra-shells (K,L) and the rest for inter-shells(KL) . The results are obtained numerically by using computer programs (Mathcad).

Excitation Energies from a Partially Spin-Restricted Wave Function

2007 ◽  
Vol 3 (1) ◽  
pp. 65-69 ◽  
Author(s):  
V.N. Glushkov
Keyword(s):  
Wave Function ◽  
Excited States ◽  
Electron Correlation ◽  
Electronic Excitation ◽  
Slater Determinant ◽  
Molecular Orbitals ◽  
Reference Configuration ◽  
Excitation Energies ◽  
Hartree Fock ◽  
Natural Base

A singe Slater determinant consisting of restricted and unrestricted, in spins, parts is proposed to construct a reference configuration for singlet excited states having the same symmetry as the ground one. A partially restricted Hartree-Fock approach is developed to derive amended equations determining the spatial molecular orbitals for singlet excited states. They present the natural base to describe the electron correlation in excited states using the wellestablished spin-annihilated perturbation theories. The efficiency of the proposed method is demonstrated by calculations of electronic excitation energies for the Be atom and LiH molecule.

CALCULATION OF THEORETICAL ISOTROPIC COMPTON PROFILE FOR MANY PARTICLE SYSTEMS

2010 ◽  
Vol 24 (14) ◽  
pp. 1601-1614
Author(s):  
ALI A. ALZUBADI ◽  
KHALIL H. ALBAYATI
Keyword(s):  
Wave Function ◽  
Impulse Approximation ◽  
Particle Systems ◽  
Electron Interaction ◽  
Compton Profile ◽  
High Momentum ◽  
Atomic Systems ◽  
Hartree Fock ◽  
Hf Wave ◽  
Momentum Region

Theoretical isotropic (spherically symmetric) Compton profiles (ICP) have been calculated for many particle systems' He , Li , Be and B atoms in their ground states. Our calculations were performed using Roothan–Hartree–Fock (RHF) wave function, HF wave function of Thakkar and re-optimized HF wave function of Clementi–Roetti, taking into account the impulse approximation. The theoretical analysis included a decomposition of the various intra and inter shells and their contributions in the total ICP. A high momentum region of up to 4 a.u. was investigated and a non-negligible tail was observed in all ICP curves. The existence of a high momentum tail was mainly due to the electron–electron interaction. The ICP for the He atom has been compared with the available experimental data and it is found that the ICP values agree very well with them. A few low order radial momentum expectation values 〈pn〉 and the total energy for these atomic systems have also been calculated and compared with their counterparts' wave functions.

On the basis of orbital theories

1955 ◽  
Vol 232 (1188) ◽  
pp. 114-135 ◽  
Keyword(s):  
Wave Function ◽  
Configuration Interaction ◽  
Singlet Ground State ◽  
New Techniques ◽  
Hartree Fock ◽  
Basic Set ◽  
Spin Eigenfunctions ◽  
Spin Degeneracy ◽  
One And Many ◽  
Reduced Density

In molecular theory the wave function is usually constructed from antisymmetrized products, or ‘Slater determinants’, of one-electron ‘orbitals’. A single determinant of suitably chosen, doubly occupied orbitals is often a fair approximation to a singlet ground state; but when more general products are admitted, as in ‘configuration interaction’ calculations, it is first necessary to resolve a high ‘spin degeneracy’ by constructing spin eigenfunctions (SE’s). In §1, the fundamental basis of recent methods (McWeeny 1954 b ) is clarified by a group theoretical approach. Next, in §2, the energy expression, using as wave function an arbitrary mixture of similar SE’s, is written very simply in terms of the reduced density matrices for one and two particles, and formulae for the calculation of these matrices are given. The remaining problem is to get a ‘best’ wave function, usually with limited configuration interaction, by (i) variation of SE coefficients and (ii) variation of the orbitals appearing in the SE’s; this problem is formally solved in §3. (i) is the usual configuration interaction process; but (ii) is new and leads, when the orbitals are expressed in terms of a standard basic set (e.g. of atomic orbitals), to a complete generalization of the Roothaan 1951) equations. These (matrix) equations are simple in appearance, but their numerical solution calls for new techniques; and it is possible that the Roothaan (i.e. Hartree–Fock) approach, followed by configuration interaction, provides about the best working compromise between (i) and (ii). In §4, some points of contact between one- and many-configuration theories are noted. In particular, certain density matrix elements provide appropriate generalizations of the ‘charges’ and ‘bond orders’ of Coulson and Longuet-Higgins and continue to describe the response of a system to changes in its ‘Coulomb’ and ‘resonance’ integrals.

An expansion method for calculating atomic properties IV. Transition probabilities

1964 ◽  
Vol 280 (1381) ◽  
pp. 258-270 ◽  
Keyword(s):  
Wave Function ◽  
Matrix Element ◽  
Transition Probabilities ◽  
Expansion Method ◽  
The State ◽  
Diagonal Matrix Element ◽  
Expectation Value ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

An earlier expression for the expectation value of a single-electron operator which isstationary with respect to first-order variations of the state wave function has been generalized to the case of an off-diagonal matrix element connecting two different states. Explicit calculations are carried out of the probabilities of dipole transitions between configurations 1 s a 2 s b 2 p c and 1 s a 2 s b–1 2 p c+1 for all members of the isoelectronic sequences from helium to neon and the importance of taking into account the mixing of degenerate configurations is demonstrated. The accuracy is at least comparable to that of the Hartree-Fock approximation and in cases where degeneracy is important it is much superior.

Transition probability approach for direct calculation of coefficients of Configuration Interaction wave function

2017 ◽  
Author(s):  
Arijit Bag
Keyword(s):  
Wave Function ◽  
Configuration Interaction ◽  
Correlation Energy ◽  
Transition Probability ◽  
Direct Calculation ◽  
Energy Curve ◽  
Equilibrium Bond Length ◽  
Hydrogen Water ◽  
Hartree Fock ◽  
A New Technique

To reduce the computation cost of Configuration Interaction (CI) method, a new technique is used to calculate the coefficients of doubly excited determinants directly from orbital energies, orbital overlap matrix and electron population obtained from Hartree Fock level run. This approach to approximate the coefficients of CI wave function is termed as <b>transition probability approximated CI (TPA-CI).</b> In principle, calculated dynamical electron correlation energy of TPA-CI and Full CI (FCI) are equivalent. It is observed that computed TPA-CI correlation energies of hydrogen, water, ammonia and ozone are very close to FCI values, within 5% error. The potential energy curve of the hydrogen molecule is also studied and it is found that the energy is minimum at its equilibrium bond length.<br><br>

Correlated Atomic Pair Functions by the e-ϱ-Method. I. Ground State 11S and Lowest Excited States n1S (n > 1) and n3S of Helium

1999 ◽  
Vol 54 (12) ◽  
pp. 711-717
Author(s):  
F. F. Seelig ◽  
G. A. Becker
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Helium Atom ◽  
Excited States ◽  
Ground State ◽  
Single Parent ◽  
Hartree Fock ◽  
Atomic Pair ◽  
Electronic Wave Function ◽  
S States

Abstract Some low n1S and n3S states of the helium atom are computed with the aid of the e-e method which formulates the electronic wave function of the 2 electrons ψ = e-e F, where ϱ=Z(r1+r2)–½r12 and here Z = 2. Both the differential and the integral equation for F contain a pseudopotential Ṽ instead of the true potential V that contrary to V is finite. For the ground state, F = 1 yields nearly the Hartree-Fock SCF accuracy, whereas a multinomial expansion in r1, r2 , r2 yields a relative error of about 10-7 . All integrals can be computed analytically and are derived from one single “parent” integral.

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.

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