Approximate Ground State Wavefunction Research Articles

Magnetic properties and spin waves of bilayer magnets in a uniform field

: The two-layer square lattice quantum antiferromagnet with spins 12 shows a zero-field magnetic order-disorder transition at a critical ratio of the inter-plane to intra-plane couplings. Adding a uniform magnetic field tunes the system to canted antiferromagnetism and eventually to a fully polarized state; similar behavior occurs for ferromagnetic intra-plane coupling. Based on a bond operator spin representation, we propose an approximate ground state wavefunction which consistently covers all phases by means of a unitary transformation. The excitations can be efficiently described as independent bosons; in the antiferromagnetic phase these reduce to the well-known spin waves, whereas they describe gapped spin-1 excitations in the singlet phase. We compute the spectra of these excitations as well as the magnetizations throughout the whole phase diagram.

Variational Principle for a Particle in a Box

The particle-in-a-box problem is reexamined, using different model wave functions, to illustrate the use of the variational principle applied to the simplest solvable quantum mechanical problem. The scheme presented here provides a useful paradigm for the LCAO approach used in atomic and molecular calculations. Suitable polynomial basis functions are introduced that provide a way for controlling both the accuracy and the number of independent variational parameters used in the energy calculations, when the coordinate origin is taken either at the left-hand edge or the center of the box. Links with the Bransden and Joachain calculations on the first two even parity states are made. The approximate ground state wave function, x(x - L), is also redefined by introducing a variation parameter in the form of a prescription based on the value of the position coordinate. Finally, a straightforward method is presented for determining an approximate energy for an arbitrary excited state, based on the use of any app...

Variational supersymmetric WKB approximation

In this paper we propose a way to get the approximate ground-state wave function by the variational method and thence determine the superpotential $w$ that is necessary for the supersymmetric WKB quantization condition. By this approach, the supersymmetric WKB method can be applied for any potentials. Two examples are given: (i) the anharmonic potential and (ii) the Hulth\'en potential including the centrifugal potential.

Maximum-entropy principle and neural networks that learn to construct approximate wave functions.

By recourse to a maximum-entropy-based method for the training of perceptrons, we show that an appropriately prepared network is able to construct approximate ground state wave functions of good quality with the sole knowledge of a few expectation values, even if nothing is known concerning the interaction potential. \textcopyright{} 1996 The American Physical Society.

WKB potential well ground states without matching?

Schmid has recently conjectured that it might be possible in general to obtain semi-classical (WKB) ground state energies and wavefunctions for various potential surfaces without explicitly considering the problem of how wavefunctions obtained separately in the classically allowed and classically forbidden regions are to be matched across the boundary between those regions. We show, by formally expanding the energy as a power series in kh in the usual WKB equations, that even in the one-dimensional case, Schmid's procedure yields an approximate ground state wavefunction only for potential wells which are nearly harmonic, and that the corresponding energy obtained is totally insensitive to any anharmonicity present. Furthermore, excited state wavefunctions cannot be obtained at all. Thus, despite its inherent simplicity, Schmid's method can only be of limited value in the analysis of the dynamics of real molecules on multi-dimensional potential surfaces.

Random phase approximation ground state correlation energy of an interacting phonon system

It is shown that the nuclear ground state phonon correlation energy can be easily calculated using an approximate ground state wavefunction in the random phase approximation. Numerical examples are reported for Ni and Zn isotopes. According to the calculation, the magnitude of this energy is about 5 MeV which is much larger than the size of the nuclear pairing energy.

Wavefunction comparisons for the valence-bond model for conjugated ?-networks

Approximate ground-state wavefunctions for valence-bond (or Heisenberg) models are obtained both within Neel-state-based and within Kekule-state-based resonance-theoretic approaches. Comparisons are made between these and other general approaches, with particular emphasis on organic π-network systems. Attention is drawn to the manner in which the quality of the different approximation schemes changes with variations in structural characteristics of the system. It is suggested that resonance-theoretic ideas are most appropriate for (aromatic benzenoid) systems with low coordination number, whereas Neel-state based ideas are most appropriate for (3-dimensional) structures with higher coordination number (and little “frustration”).

Analysis of approximations and errors in equations of motion method calculations

We analyze a number of fundamental questions associated with the use of a finite one-particle orbital basis in equations of motion (EOM) method calculations of excitation energies etc., of atomic and molecular systems. This approximation yields an approximate ne-electron ground state and say, N excited states, while there are (N + 1)2 different possible basis operators for EOM calculations. We show that sets of at most 2N basis operators can contribute to the EOM calculations. Any set of 2N basis operators, satisfying certain conditions, provides the exact EOM energies which are equivalent to complete configuration interaction results within the same orbital basis. We investigate the use of particle-particle shifting operators which are not employed in EOM calculations in model calculations on He with operator bases smaller than the complete 2V to consider the convergence of the expansion. The dependence of EOM calculations on the quality of the approximate ground state wavefunction is studied through calculations for Be where additional support is provided for the frequent need for multiconfigurational zeroth order reference functions (as corrected perturbatively). Excited state EOM wavefunctions from EOM calculations are shown to not necessarily be orthogonal to either the exact or approximate ground state wavefunction, suggesting implications in the use of EOM methods to evaluate excited state properties. The He and Be examples and a simple two-level problem are also utilized to illustrate questions concerning the use of the EOM equations to obtain an iteratively improved ground state wavefunction.

Momentum Representation of Hydrogenic Ground States Derived from Complicated Band Extrema

AbstractA simple structure of an approximate ground state wave function is suggested in the case of a particle with complicated effective mass bound by a Coulomb potential. The state is expressed in the momentum representation and contains only the binding energy and the explicit band structure near the band extremum. The theory is checked by investigating separately two cases: ellipsoidal, parabolic bands and spherical, non‐parabolic bands.

Ground states and photoionization of iron group transition metal impurities in semiconductors

Approximate ground state wavefunctions and potentials for iron group transition metal impurity centres in semiconductors, for instance Cr, Fe, Cu, Ni, Co, Mn in GaAs are calculated using a model in which both the screened long range Coulomb potential and the short range core potential are taken into consideration. The magnitude and the wavelength dependence of the photoionization cross section are obtained via the wavefunction, in favourable agreement with experiment.

Study of Different Approximation in the Calculation of g Tensors: H2+

The theory of the g tensor in one-electron systems is briefly reviewed and calculations are done in several ways for the H2+ molecular ion in order to test approximations which must be made for larger systems. Approximate ground state wavefunctions are determined variationally using a power series in elliptic coordinates and as a linear combination of atomic orbitals. The first-order wavefunction with respect to the orbit-field perturbation is calculated variationally and by a Rayleigh–Schrödinger expansion and the second-order g tensor determined. Excited state functions and energies are approximated from the higher roots of the secular equations involved. The results of the various approximate calculations are compared and discussed. It is found that the LCAO method is rather poor and that two center integrals cannot be neglected. It also appears that a single spin–orbit parameter cannot be used for different orbitals on the same atom. The best result, obtained from the variationally determined first-order function, has gxx differing from the free electron value by −0.225 × 10−5.

Magnetic and Electrical Properties of Small Diatomic Molecules. I. Diamagnetic Susceptibility, Proton-Shielding Constant, and Electric Polarizability of the Hydrogen Molecule

An unambiguous exact variational procedure is given for the calculation of certain weak interaction energies starting from any given approximate ground-state wavefunction. The method is illustrated by numerous calculations on the hydrogen molecule using five different ground-state wavefunctions.

Phenomenological two-nucleon potentials and the deuteron, II

The ground state of the deuteron corresponding to Gammel and Thaler's potential has been investigated by numerical integration. There exists no exact ground state in the neighbourhood of the right binding energy. An approximate ground state wave function has been tabulated and the corresponding electromagnetic properties of the deuteron have been studied, including the correction to the magnetic moment due to the LS- potential.

Phenomenological two-nucleon potentials and the deuteron

Abstract The ground state of the deuteron corresponding to Gammel and Thaler's potential has been investigated by numerical integration. There exists no exact ground state in the neighbourhood of the right binding energy. An approximate ground state wave function has been tabulated and the corresponding electromagnetic properties of the deuteron have been studied, including the correction to the magnetic moment due to the LS- potential.

A Comparative Survey of Approximate Ground State Wave Functions of Helium Atom and Hydrogen Molecule

A Comparative Survey of Approximate Ground State Wave Functions of Helium Atom and Hydrogen Molecule

Criteria of Goodness for Approximate Wave Functions

The relations between the root-mean-square error of an approximate ground state wave function, the energy error, and the root-mean-square local energy deviation are exhibited and discussed. Computations of these quantities for helium wave functions containing errors of different character and magnitude are presented, and are shown to indicate that the errors in wave functions of the Hylleraas type are of increasingly short range character as these functions are made more flexible. The form of variational process which will give a wave function most satisfactory for a given purpose is discussed with illustrative computations relating to the diamagnetic susceptibility of He. It is found that the energy error associated with a wave function may be a comparatively unsatisfactory criterion of goodness.