Everett G. Larson

Study of Electron Correlation in the H6 Ring, Using a Novel Approximation

Studies on electron behavior in finite model systems provide information relevant to electrons in lattices. Approximate schemes for testing electron correlation may be tested by such model calculations. We report here the results of a study on the 1Γ1 ground state of the H6 hexagonal ring, using a novel type of wavefunction containing both valence bond and molecular orbital components. The method has a number of significant advantages and gives results as good as those of the simple alternant molecular orbital (AMO) method. Possible implications of these results for molecular and solid‐state electron behavior are discussed briefly.

The Evolution of our Probability Image for the Spin Orientation of a Spin — 1/2 — Ensemble as Measurements are Made on Several Members of the Ensemble — Connections with Information Theory and Bayesian Statistics

An analysis is made of the information acquired when measurements are made, of the component of spin along an arbitrarily chosen axis, (Θ, Φ), of each of several or many spin 1/2 systems, each prepared in the same way. Since the probability p↑ of the observed value of the spin being “up” is a unique function of the relative orientation of the axis of measurement, (Θ, Φ), and the quantization axis, (θ, φ), of the state into which the system was prepared, this probability function, p↑(θ, φ; Θ, Φ), may be interpreted as a likelihood function for the orientation, (θ, φ), of the state into which the system is prepared, conditioned by its having been measured with “up” spin along the axis with orientation (Θ, Φ). If one assumes that the preparation procedure prepares each system into the same unknown pure state, and further assumes that, a priori, each equal infinitisimal solid angle is equally likely to contain the orientation of this pure state, the likelihood function for the orientation, via Bayes’ theorem, becomes the (unnormalized) probability density function for the orientation of the pure state. Thus, from an ensemble of spin 1/2 systems, each prepared in the pure state with quantization axis (θ, φ), the probability for realizing n “up” values and m = N—n “down” values, in some specified order, in a sequence of N measurement events for the spin component along the (Θ, Φ) axis, is given by Π(θ, φ; Θ, Φ) = p ↑ n(1-p ↑)m = p ↑ n(1-p ↑]) N-n , (as in the Bernoulli coin-tossing problem). By the above arguments Π(θ, φ; Θ, Φ) dΩ, becomes, for fixed (Θ, Φ,n,N), the (unnormalized) probability for the system being prepared in the spin state whose direction is aligned to within dΩ of (θ, φ), conditioned by the measurement events which determined Π. If we take the density operator corresponding to this distribution function to be ρ = [∫dΩ,ρ0(θ, φ)Π(θ, φ; Θ, Φ)]/[∫dΩΠ(θ, φ; Θ, Φ)], where ρ0(θ, φ) is the density operator corresponding to the pure state aligned along (θ, φ), one finds that the probability of measuring “up” spin along (Θ, Φ), conditioned by a measurement of n “up” values in N trials is given by (n+1)/(N+2), which is Laplace’s rule of succession. Also presented is a geometrical parametrization of the density operator for a spin 1/2 system. The density operator for the pure state with spin orientation (θ, φ) is represented as the corresponding point on the unit sphere. Each point in the interior of the sphere represents a non-idempotent density operator, diagonal in the coordinate system in which the radial line to the representative point lies on the +2 axis, and such that the radius is equal to (σz) = λ↑ — λ↓. A measurement of n “up” values in TV trials, with a prior probability distribution that is uniform over the volume of this sphere, leads to (n+2)/(N+4) for the probability of measuring “up” spin; a more “cautious” rule of succession than that of Laplace.

Oscillator strengths for the 2s2, 2p2, 2s3s(1S)-2s2p(1P) transitions of B(II)

Abstract Accurate configuration interaction wavefunctions for the 2 s 2 ( 1 S ), 2 p 2 ( 1 S ), 2 s 3 s ( 1 S ) and 2 s 2 p ( 1 P ) states of B(II) are calculated in a single optimized orbital basis of 7 s , 6 p and 4 d Slater-type orbitals. 95, 84, 57 and 90% of the correlation energies, respectively, are realized by these wavefunctions. Oscillator strengths for the three 1 S - 1 P transitions are calculated from these and from less accurate wavefunctions in the same orbital basis. The length values obtained from our most accurate wavefunctions, in order of increasing 1 S energy, are 0.9885, 0.202 and 0.007. The degree of accuracy of these oscillator strengths is estimated by noting the convergence to final values as increasing percentages of correlation energies are included in the wavefunctions together with the increasing agreement between length and velocity formulas. The exact theoretical oscillator strength for the resonant line is projected to be 0.985 with an error almost certainly not greater than ±0.015. The theoretical oscillator strengths for the other lines are considered to be 0.21±0.02 and less than 0.007, respectively.

Bayesian Logic and Statistical Mechanics — Illustrated by a Quantum Spin 1/2 Ensemble

The greatest challenge for the appropriate use of Bayesian logic lies in the need for assigning a “prior”[1–13]. In physical applications, this prior is often strongly molded by the physical “boundary conditions”[1,12,13], and thereby begins to take on some aspects of “objective” meaning. Many claim that the prior usually plays only a minor role in the interpretation of the data. We shall show situations in which a prior, objectively justified by the physical constraints, can lead to a very strong molding of our interpretation of the measured data. We shall also show how the concept of a “probability distribution function over probability distribution functions”[8,9] has practical utility in clarifying and taking advantage of the distinction between the “randomness” intrinsic in the preparation of an ensemble of physical systems, and the uncertainty in our limited knowledge of the values of the parameters that characterize this preparation procedure. In this paper we consider three physical situations in which Bayesian concepts lead to a useful, non-conventional interpretation of the information contained in a small number of physical measurements. The first two of these relate to an ensemble of quantum mechanical spin-1/2 systems, each prepared by the same procedure[13]. The third relates to the energies of molecules sampled from a fluid that obeys the classical statistical mechanics of Gibbs[1,8,9]. In the first example, we show how a prior chosen to match the physical constraints on the system leads to a very conservative “rule of succession”[7,10,3] that discounts any significant deviation from “random” frequencies (for measured values) as being nonrepresentational of the distribution being sampled. In the second example, we show how an appropriate, physically motivated prior leads to a rule of succession that is less conservative than that of Laplace[5], and even, in most circumstances, is less conservative than the maximum likelihood inference[1–3] from the observed frequency. This example also shows how a prior that is uniform in one parameter space can yield a rather peaked prior in the space of another related (and physically important) parameter, (in this case, the temperature). In the third example, we show how Bayesian logic together with the knowledge of the results of only a small number of physical measurement events augments equilibrium statistical mechanics with non-Maxwellian distribution functions and probability distribution functions over temperature. These distribution functions are a more faithful representation of our state of knowledge of the system than is a Maxwellian distribution with a temperature inferred by the principle of maximum likelihood [1–3,14,9].

REDUCED DENSITY OPERATORS, THEIR RELATED von NEUMANN DENSITY OPERATORS, CLOSE COUSINS OF THESE, AND THEIR PHYSICAL INTERPRETATION

The (N-l)- and (N-2)-electron reduced density operators of a pure stationary state of a many electron system (such as the electronic structure of an atom or molecule) can be realized as an idealization of the statistical von Neumann density operator of the ion formed from that pure stationary state in an appropriate sudden ionization experiment. The N-electron Hermitian operator whose matrix representation is the Garrod and Percus G-Matrix (the metric matrix of the particle-hole states formed from the given pure stationary state) can be realized as the excitation-related part of the statistical von Neumann density operator of the system formed from this pure stationary state via a sudden random impulsive particle-conserving perturbation. Comparison of these statistical von Neumann density operators with those obtained from specified perturbations allows discrimination between responses of the system which depend strongly upon the excitation mechanism and those which are characteristic of the system and only weakly dependent on the excitation mechanism.

Computational Exploration of the Entropic Prior Over Spaces of Low Dimensionality

Bayesian methods provide an objective analysis for problems with incomplete information. Yet, the use of Bayesian methods requires the assigning of an a priori probability, or prior. The prior should be assigned to contain the least information, while at the same time being consistent with the statistical parameters of the problem. To do this correctly, a complete integration of Information Theory with Bayesian Estimation methods is necessary. When finding probability distributions over the probability assignments, traditional methods are not entirely self-consistent. In papers presented at the 1996 MaxEnt Workshop, Larson, Evenson, and Dukes demonstrated that commonly used methods minimize only part of the total information. Minimizing the total information produces an entropic prior. Implementing this complete method to find the best probability distribution over probability assignments for three- to five-sided dice showed that the entropic prior Bayesian method gives results which differ significantly from these standard approaches. This is especially apparent at low dimensions.

A Lognormal State of Knowledge

The Lognormal distribution is derived as the representation of a particular state of knowledge using a combination of maximum entropy and group invariance arguments.

On the Assignment of Prior Expectation Values and a Geometric Means of Maximizing — Trρln ρ Constrained by Measured Expectation Values

An alternative method for applying the maximum entropy criterion to a simple two state system is presented. It evolves only a few rudiments of geometry and group theory. The advantages of this method are a vivid geometric depiction of the maximum entropy state as well as its analytic solution. The relationship between the maximum entropy state and the assignment of a prior expectation value made with it is also made apparent.