Correlated complex exponential wave functions for atomic-molecular systems

The application of flexible correlated complex exponential basis functions depending on all inter-particle distances in precision variational computations of atomic-molecular systems is considered. In contrast to three-particle systems, the application of such functions for four-particle systems is hindered by the cumber-some nature of the multistep algorithm of evaluation of the integrals determining the matrix elements of the energy operator. Transformations reducing the number of integrals to be evaluated are carried out. As a result, the computation volume is reduced by a factor of six: instead of 43 integrals, it is sufficient to take 6 integrals of the Coulomb interaction of pairs of particles and the overlap integral. The method of direct construction of formulas for many-particle integrals is discussed.     

Integral Hellmann-Feynman theorem as a criterion of the quality of wave functions of atomic-molecular systems

It is proposed to use the integral Hellmann-Feynman theorem for the quality control and refinement of atomic and molecular wave functions. Its validity is verified for variational wave functions of the positronium ion e ⁺ e ^? e ^?, the negative ion of the hydrogen atom p _∞ ⁺ e ^? e ^? (H^?), and mesomolecules μ^?π⁺π⁺ and μ^? p ⁺ p ⁺. The relative violation of this theorem (10^?2) is six orders of magnitude larger than the error of the energy calculation (10^?8), which demonstrates its high sensitivity to the quality of wave functions. A way of refining wave functions on the basis of combination of the integral Hellmann-Feynman theorem for exactly solvable model and real atomic-molecular systems is proposed. A rule for verification of the mutual consistency of the wave functions of any three quantum-mechanical systems is formulated.     

Partial virial theorems for atomic-molecular systems

New virial relations for three-and four-particle atomic-molecular systems are proposed. Using operators of extension or squeezing of interparticle distances, it is shown that, for all pairs of j and k particles in S states of these systems, the following partial virial relations are valid: 〈2T _jk 〉+〈 V _jk 〉=0, where 〈V _jk 〉 is the average Coulomb interaction energy for a pair of particles and 〈T _jk 〉 is a part of the average kinetic energy of the system. There are three and six such relations for three-and four-particle systems, respectively. The conventional virial theorem (〈 2T〉+〈V〉=0) for the average total kinetic and potential energies of the system (〈 T〉 and 〈V〉, respectively) corresponds to the summation of partial virial relations over all pairs of particles. It is shown by an example of variational calculations of the helium atom ⁴He²⁺ e ^? e ^? and the helium muon-electron mesoatom ⁴He²⁺μ^? e ^? that partial virial relations are a highly sensitive indicator of the accuracy of wave functions.     

Exponential decay of bound state wave functions

The spatial decay properties of the wave functions of multiparticle systems are investigated. The particles interact through pair potentials in the classR+L . The bound states lie below the bottom of the continuous spectrum of the system. Exponential decay, in anL ² sense, is proven for these wave functions. The result is the best possible one which will cover every potential in this class.Based on a thesis submitted to Princeton University in partial fulfillment of the degree of Doctor of Philosophy.     

Partial hypervirial theorems for atomic-molecular systems

A family of partial hypervirial theorems for physical properties of pairs of particles in three-and four-particle atomic-molecular systems is considered. The partial hypervirial theorems generalize the partial virial theorems proposed earlier. The sum rule is formulated, according to which the expectation values of the derivatives with respect to the interparticle distances are related to the products of the charges of pairs of the particles. This rule is used to check the accuracy of the calculation of variational wave functions for the positronium ion e ^? e ^? e ⁺. It is shown that the sum rule is two orders of magnitude more sensitive to the inaccuracy of the calculation of the wave function than the partial virial theorems and is five or six orders of magnitude more sensitive to it than the conventional virial theorem.     

Exact wave functions for concentric two-electron systems

We show that the exact solution of the Schrödinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular values of the ring or sphere radii. In the case of rings, we report exact polynomial and irrational solutions. In the case of spheres, we report exact polynomial solutions for the ground and excited states of S symmetry.     

Exponential wave maps

We generalize wave maps to exponential wave maps. We compute the first and second variations of the exponential energy, and obtain results concerning the stability of exponential wave maps. We prove a theorem which relates wave maps, exponential wave maps, and the conservation law of second-order symmetric tensors. We show that if f is an exponential wave and a pseudo-weakly conformal map, then f is homothetic. We finally discuss the applications of exponential wave maps in relativity.     

Integrals in the theory of four-particle atomic-molecular systems

Explicit expressions for four-particle Coulomb interaction integrals and overlap integrals for exponential basis functions dependent on five interparticle distances have been obtained. These formulas significantly simplify calculations in comparison with the general algorithm and are free from uncertainties arising in its application. The values of the integrals under consideration are calculated for a number of points forbidden in the general algorithm. The results obtained are applicable to analysis of four-particle integrals of the general form and calculation of four-particle atomic-molecular systems with partial allowance for the correlation of particle motion.     

Resonating valence bond wave functions for strongly frustrated spin systems

The resonating-valence-bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration." This scenario, proposed a long time ago but never confirmed by microscopic calculations, is strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J(1)-J(2) Heisenberg model for 0.4 less than approximately J(2)/J(1) less than approximately 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.     

Monte Carlo optimisation of correlated wave functions for normal two-electron systems

A simple, new type of correlated wave function is proposed for the studies of normal two-electron atomic systems: ψ(r₁, r₂) = Σc_mΦ_m(r₁, r₂) with Φ_m(r₁, r₂) = exp[−(r₁ + r₂)]/(br₁₂ + a)^m, where , a, b are non-linear variational parameters. A notable feature of this basis function is that only three terms are required within the framework of the Raleigh-Ritz variational principle to obtain fairly accurate energy eigenvalues and satisfactory cusp conditions. The non-linear variational parameters are optimised by using the Monte Carlo technique.     

Compact variational wave functions for bound states in three-electron atomic systems

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r ₁₂, r ₁₃, r ₂₃, r ₁₄, r ₂₄, and r ₃₄. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 1² S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be⁺, B²⁺, C³⁺, ..., Na⁸⁺, and Mg⁹⁺. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q ⁻¹ expansion are derived for the total energies of these three-electron systems. The article is published in the original.     

Relation between Bethe-Salpeter and Schrödinger wave functions for many-particel systems

The connection is made between a many-time approach to S-matrix elements and energy eigenvalues, which naturally arises from a field theoretical point of view, and the single time Schrödinger- and Breit-like formalism often used in detailed calculations for many-particle systems, such as many-electron atoms. Specifically, the many-particle Bethe-Salpeter equation is expressed in terms of the corresponding Schrödinger equation for the non-relativistic case in which the Bethe-Salpeter kernel consists of only two-particle local static interactions. Also, the one-photon transition matrix element for this case in the Bethe-Salpeter formalism is shown to be equivalent to the corresponding well-known Schrödinger result. The treatment developed is well suited to systematic relativistic generalization.     

Exponential clustering for long-range integer-spin systems

By using Kirkwood-Salsburg equations for classical spin systems with unbounded integer values we prove exponential decay (resp. power law decay) for exponential (resp. power law) decaying potentials. We use these results to prove the mass gap in the two-dimensional Higgs-Villain model in the weak coupling region.Research partially supported by the National Science Foundation under Grant PHY-77-18762On leave from Department of Mathematics, University of British Columbia, Vancouver, B. C., Canada. Research partially supported by National Research Council under Grant A4015On leave from Istituto di Fisica Teorica, Napoli, Italy     

Criterion for the accuracy of wave functions of asymmetric atomic and molecular systems

From the condition for invariance of the energy under rotations of the Jacobi coordinates, a relationship is obtained that relates the expected values of combinations of degrees of interparticle separations in quantum-mechanical systems. This relationship, which is satisfied for atoms and molecules with like particles identically, because of the permutational symmetry, is an extremely sensitive criterion for the accuracy of the wave functions of asymmetric systems, consisting of unlike particles and having no permutational symmetry. The application of this criterion is illustrated for the example of the asymmetric ⁴Heμ^?e^? mesoatom.     

Factorization of wave functions of the quantum integrable particle systems

The relation between the characteristics of the equilibrium configurations of the classical Calogero-Moser integrable systems and properties of the ground state of their quantum analogs is found. It is shown that under the condition of factorization of the wave function of these systems the coordinates of classical particles at equilibrium are zeroes of the polynomial solutions of the second-order linear differential equation. It turns out that, under these conditions, the dependence of classical and quantum minimal energies on the parameters of the interaction potential is the same.