Wave Function Of System Research Articles

Alternate analysis of Koç's thought-experiment

Koç's thought-experiment is re-analyzed, keeping track of the total system wavefunction. It is shown that there is no contradiction with the usual postulates of quantum mechanics.

A resolvent approach towards the electronic structure of systems constructed of chain fragments. Branched polyenes

A resolvent approach (in the HMO approximation) towards the electronic spectra and wavefunctions of systems consisting of chain fragments has been proposed. As an application, branched polyene chains with/without bond alternation have been considered. It is shown that the branching of an infinite polyene leads to the appearance of local states, whose energies and wavefunctions have been studied extensively.

A variation perturbation calculation to study the excited-state wavefunctions of atoms

A variation perturbation calculation using coupled Hartree-Fock (CHF) theory has been presented to study the excited-state wavefunctions of many-electron systems. The excited-state wavefunctions are extracted from the poles of an appropriate functional obtained from a linearised time-dependent response of the system under external frequency-dependent perturbation. The theory has been applied to find the excited n1S (n>1) functions of two-electron atomic systems. Excellent agreement is observed for the transition energies and matrix elements of different operators.

A Comparison of Mesquac- and “Full” AB Initio MO-Wavefunctions

The quality of wavefunctions for complex systems derived by either “full” ab initio MO-SCF calculations or within the MESQUAC-MO framework [1] is investigated comparing chemically interesting quantities as atomic and overlap populations and the character of the chemical bond. A direct relation between the results of both methods is shown to exist, allowing an extrapolation of the much less time consuming MESQUAC computations to the “full” MO SCF level. Hydrate complexes of main group and transition metal ions have been chosen for some practical applications

SCF CI calculations for vibrational eigenvalues and wavefunctions of systems exhibiting fermi resonance

It is shown that configuration-mixing calculations with configurations built from SCF wavefunctions can be used to provide accurate and compact wavefunctions for systems in which the independent-mode approximation breaks down.

Spherical gaussian basis sets in relativistic quantum chemistry

A formalism has been developed so that spherical gaussian basis sets can be used for the calculation of relativistic Hartree-Fock wavefunctions for atomic systems. Results are presented for the point nucleus Xe atom.

A new coupled-state equation for ion-atom collisions

A new coupled-state expansion for the system wavefunction in an ion-atom collision problem is presented in the form of an integral equation. This new approach has the property that it is not unitary in the sense that the norm of the wavefunction is not equal to unity at all times. It thus allows ionised electrons to leave the target region, as they should but are not allowed to in conventional approaches. Although this problem with conventional coupled-state expansions is present in both one- and two-centred expansions, the new approach applies specifically to the case of one-centred (target-centred) expansions of the wavefunction. With this approach may it be possible to carry out efficiently coupled-state calculations in a target-centred finite Hilbert basis that will remain accurate even when ionisation is dominant.

Expansion methods for Adams–Gilbert equations. I. Modified Adams–Gilbert equation and common and fluctuating basis sets

Adams–Gilbert (AG) equation for nonorthogonal localized orbitals of a single-determinant wavefunction has been modified so as to enable one to compute wavefunctions of large polyatomic systems by the expansion method. This equation is named as modified Adams–Gilbert (MAG) equation. One solves the AG or the MAG equation by each subsystem and, collecting all the orbitals obtained, one constructs wavefunction of the system. It is shown that when one employs the expansion method, one must actually use basis functions common to all the subsystems (common basis set) to solve the AG equation, while one can employ, by each subsystem, different basis functions appropriate to the subsystem (fluctuating basis set) to solve the MAG equation. An expansion method suitable for solving the AG and the MAG equations has been presented. Application of the method to HF, H2O, and CH4 has revealed that (1) the method proposed is workable, (2) actually so many basis functions are not needed for describing some subsystems, especially for core electrons, and (3) it is necessary to orthogonalize approximately, not necessarily rigorously, the orbitals in the system.

Projected general spin-orbital wavefunctions for three-electron systems

Spin-projected Hartree-Fock calculations using general spin orbitals are performed on some S, P, D and F states of Li and on the ground states of its isoelectric ions with Z=4-10. The energy -7.447596 au is obtained for Li 22S, compared to -7.447565 au with the spin-optimized SCF method and -7.432727 au with the ordinary restricted Hartree-Fock method. Full configuration-interaction results for the same basis sets are also reported.

Relativistic self-consistent-field (RSCF) theory for closed-shell molecules

The relativistic Hartree–Fock–Roothaan (RHFR) formalism for closed-shell molecules is given. The wavefunction for such systems is taken as a single Slater determinant of 4-component molecular spinors (MS), where each MS is written as a linear combination of atomic spinors (LCAS/MS). The radial part of the atomic spinor (AS) is expanded in terms of Slater-type basis functions (STBF). The relativistic electronic Hamiltonian for the molecular system (in Born–Oppenheimer approximation) is the sum of Dirac Hamiltonians plus the interelectronic Coulomb repulsion and the magnetic part of the Breit interaction, but the retardation term is neglected at present. The reduction of the matrix elements of the relativistic Hamiltonian in terms of the nonrelativistic-type matrix elements is shown for any molecular system. Expressions for the matrix elements of the above-mentioned relativistic Hamiltonian are given for diatomics.

New-type correlated wavefunctions for three-body systems with Coulomb interaction

Simple correlated wavefunctions are reported for the ground states of three selected Coulombic systems, namely (1) the positronium negative ion e−e+e−, (2) the hydrogen negative ion H−, and (3) the mesic molecular ion pμp. They give quite accurate energy results, and are analytically simple enough to be used widely in applications. Two characteristic features of the proposed wavefunctions are that (1) the correlation between the two particles with the same sign of the charge is explicitly included in the form G= xn3e−γx3, and (2) the variational parameters to be optimized are mostly nonlinear (exponential).

Perturbation Treatment for Two-Particle Symmetrized Systems with Interactions

Perturbation of a simple two particle system is briefly reviewed in one dimension leading to symmetrization of the total system's wave function for the case of identical non-interacting particles. The results of admitting some mutual particle interactions are studied by observing changes in the total system's wave function. For the case of hard-core interactions the effects on the system's symmetry properties are computed.

On absolute cross sections for two-particle transfer between heavy ions

Ground state wavefunctions for systems of two particles outside an inert core are calculated by diagonalizing a delta interaction in a Sturmian basis. The wavefunctions are used in calculations of cross sections for two-particle transfer reactions between heavy ions. The magnitude of the cross section is shown to depend strongly on the number of components of the basis included to describe the overlap.

How to resolve the orbital ambiguity to obtain the orbital set which is stable to an excitation

We discussed how the orbital ambiguity in general SCF theory can be removed to obtain the orbital set which is stable to an excitation. The orbital ambiguity for the ground state wavefunction can be resolved while at the same time performing the configuration interaction calculation for the specific singly excited states. That is, the orbitals can be uniquely determined so that they have the physical significance that they satisfy particular types of the Brillouin conditions for the excited states considered in addition to the Brillouin theorem for the ground state. This implies that the definition of Huzinaga's potential to obtain meaningful virtual orbitals is generalized to include wavefunction for open-shell systems.

Radial correlation in atoms

A method is described to calculate correlation energy in atoms. The total wavefunction of ann-electron system is expressed as a linear combination of products ofn one-electron basis orbitals. This function gives a correlated description of the system. Under suitable restrictions it reduces to DODS and to splitshell description of closed-shell atoms or molecules. Energies of He atom, He-like ions and also of H− ion have been calculated including radial correlation only. The calculated electron affinity of H is better than earlier split-shell calculations. The result for He shows that the energy limit for radial correlation has been attained. For the other 2-electron ions radial correlation alone explains about one-third of the total correlation.

Non-Hermitian representations in localized orbital theories

We consider a non-Hermitian version of Adams' localized orbital equations which is directly analogous to the conventional pseudopotential equation of Austin, Heine, and Sham. This allows the use of the familiar ``pseudizing'' arguments in localized orbital theories and also permits a simple discussion of the relationship between Adams' theory and Anderson's self-consistent pseudopotential theory of localized orbitals. A study of the differences in the Hermitian and non-Hermitian representations shows that the non-Hermitian localized orbital equation has additional localization of the left (adjoint) eigenfunction which may have advantages in practical calculations. An analysis of the secular equation method of going from the localized orbitals to the total system's wavefunction suggests that a non-Hermitian representation again has additional local properties which could prove useful in interpreting and extending semiempirical parameterization methods such as the Hückel theory.

Correlated one-center wavefunctions for two-electron molecules

A one-center configuration-interaction wavefunction for two-electron systems, built from nonorthogonal exponential-type orbitals may be multiplied by the correlation factor 1+αr12 and subjected to a variational treatment. All integrals can be evaluated, and are given in closed form. Applications are made to the ground state and an excited 1σ+ state of HeH+. The uncorrelated wavefunctions are based on the work by Stuart and Matsen. Due to the correlation factor the ground state energy at R=1.4 a.u. improves by 2.3% for a one-term and by 0.49% for a twenty-term wavefunction. The optimized value of α decreases as the number of terms increases. The best energy obtained for the ground state is −2.97458 a.u. at R=1.4 a.u. with α=0.27. Energy improvements for different R and changes in the orbital exponents were also studied. For the excited state the energy lowering due to the correlation factor becomes insignificant as the number of terms increases.

Time-Dependent Perturbation Theory of the Hartree—Fock Wavefunction of Atomic and Molecular Systems

It is shown that time-dependent perturbation theory can be applied exactly to the Hartree—Fock model of closed-shell atoms and molecules by means of integration over the excited-state continuum. The frequency-dependent polarizability of the Be atom is calculated as an example.

The exclusion priniciple and aperiodic systems

Section I .—The exclusion principle of Pauli was introduced into the old quantum theory as an empirical fact that had been brought to light in the ordering of spectra. The new mechanics has provided some sort of explanation of the principle, for in a closed system with many electrons the mathematically possible stationary states can be separated into a number of groups, with the property that transitions between stationary states in different groups cannot occur. One of these group is made up of the stationary states corresponding occur. One of the these groups is made up of the stationary states corresponding to wave functions that are antisymmetrical in the co-ordinates of the electrons. Apart from accidental degeneracies, the stationary states in different groups have different energy values. The exclusion principle then states that only energy values belonging to the antisymmertical group are found in nature. The exclusion principle has had great success not only in explaining the spectra of helium and of more complicated atoms, but also, under the form of the Fermi-Dirac statistics, in accounting for metallic conduction and ferromagnetism. In all these phenomena we are dealing with systems is stationary states, possessing energy values which are discrete, although they may lie very close together. Now, as was first emphasised by Oppenheimer, we must also use antisymmetrical wave-functions to describe aperiodic phenomena, such as the collision between an electron and an atom. If we do not, we obtain probabilities for the formation of atoms whose wave-funtions are not antisymmetrical, as we shall show in section 4, where we consider the collision between an electron and a helium atom. A helium atom described by a symmetrical wave-function would show a singlet series in palce of the observed triplets and triplet series in place of the observed singlets. The wave-functions of open systems are essentially degenerate; the symmetical and antisymmetrical solution are not separated from one another by a finite energy difference; but for any arbitrary value of the energy (and of the other integrals of the motion) we can form a symmetrical and an antisymmetrical solution. This is somewhat fundamental difference between open and closed systems. For closed systems containing two electrons there exist only the symmetrical and the antisymmetical solution; but for open systems we might take any combination of the two. In fact, to describe an observable phenomenon such as a collision, the wave-function that it would first occur to us to use is a combination of the two. To fix our ideas we shall discuss the collision between two electrons. Our arguments could equally well be applied to the collision between an electron and a hydrogen atom, the problem originally discussed by Born; but the former is the simpler case, and perhaps illustrates our theory better.