A simple and efficient approach to the optimization of correlated wave functions

We present a simple and efficient method to optimize within energy minimization the determinantal component of the many-body wave functions commonly used in quantum Monte Carlo calculations. The approach obtains the optimal wave function as an approximate perturbative solution of an effective Hamiltonian iteratively constructed via Monte Carlo sampling. The effectiveness of the method as well as its ability to substantially improve the accuracy of quantum Monte Carlo calculations is demonstrated by optimizing a large number of parameters for the ground state of acetone and the difficult case of the $1{}^1{B}_{1u}$ state of hexatriene.Comment: 5 pages, 1 figur

A Comprehensive Analysis in Terms of Molecule-Intrinsic, Quasi-Atomic Orbitals. II. Strongly Correlated MCSCF Wave Functions

A methodology is developed for the quantitative identification of the quasi-atomic orbitals that are embedded in a strongly correlated molecular wave function. The wave function is presumed to be generated from configurations in an internal orbital space whose dimension is equal to (or slightly larger) than that of the molecular minimal basis set. The quasi-atomic orbitals are found to have large overlaps with corresponding orbitals on the free atoms. They separate into bonding and nonbonding orbitals. From the bonding quasi-atomic orbitals, localized bonding and antibonding molecular orbitals are formed. The resolution of molecular density matrices in terms of these orbitals furnishes a basis for analyzing the interatomic bonding patterns in molecules and the changes in these bonding patterns along reaction paths. A new bond strength measure, the kinetic bond order, is introduced.Reprinted (adapted) with permission from Journal of Physical Chemistry A 119 (2015): 10360, doi:10.1021/acs.jpca.5b03399. Copyright 2015 American Chemical Society.</p

Effective masses for zigzag nanotubes in magnetic fields

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We obtain identities and a priori estimates in terms of effective masses and gap lengths

Accurate Ab Initio Potential Energy Curve of F2. III. The Vibration Rotation Spectrum

An analytical expression is found for the accurate ab initiopotential energy curve of the fluorine molecule that has been determined in the preceding two papers. With it, the vibrational and rotational energy levels of F2 are calculated using the discrete variable representation. The comparison of this theoretical spectrum with the experimental spectrum, which had been measured earlier using high-resolution electronic spectroscopy, yields a mean absolute deviation of about 5cm−1 over the 22 levels. The dissociation energy with respect to the lowest vibrational energy is calculated within 30cm−1 of the experimental value of 12953±8cm−1. The reported agreement of the theoretical spectrum and dissociation energy with experiment is contingent upon the inclusion of the effects of core-generated electron correlation,spin-orbit coupling, and scalar relativity. The Dunham analysis [Phys. Rev.41, 721 (1932)] of the spectrum is found to be very accurate. New values are given for the spectroscopic constants

Bound states in point-interaction star-graphs

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight "polymer", but in addition there are isolated eigenvalues unless N=2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically.Comment: 17 pages, LaTeX 2e with 9 eps figure

Alkali and Alkaline Earth Metal Compounds: Core-Valence Basis Sets and Importance of Subvalence Correlation

Core-valence basis sets for the alkali and alkaline earth metals Li, Be, Na, Mg, K, and Ca are proposed. The basis sets are validated by calculating spectroscopic constants of a variety of diatomic molecules involving these elements. Neglect of $(3s,3p)$ correlation in K and Ca compounds will lead to erratic results at best, and chemically nonsensical ones if chalcogens or halogens are present. The addition of low-exponent $p$ functions to the K and Ca basis sets is essential for smooth convergence of molecular properties. Inclusion of inner-shell correlation is important for accurate spectroscopic constants and binding energies of all the compounds. In basis set extrapolation/convergence calculations, the explicit inclusion of alkali and alkaline earth metal subvalence correlation at all steps is essential for K and Ca, strongly recommended for Na, and optional for Li and Mg, while in Be compounds, an additive treatment in a separate `core correlation' step is probably sufficient. Consideration of $(1s)$ inner-shell correlation energy in first-row elements requires inclusion of $(2s,2p)$ `deep core' correlation energy in K and Ca for consistency. The latter requires special CCV$n$Z `deep core correlation' basis sets. For compounds involving Ca bound to electronegative elements, additional $d$ functions in the basis set are strongly recommended. For optimal basis set convergence in such cases, we suggest the sequence CV(D+3d)Z, CV(T+2d)Z, CV(Q+$d$)Z, and CV5Z on calcium.Comment: Molecular Physics, in press (W. G. Richards issue); supplementary material (basis sets in G98 and MOLPRO formats) available at http://theochem.weizmann.ac.il/web/papers/group12.htm

Weakly coupled states on branching graphs

We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $\,x^{-1-\epsilon}$ along each of them, is non--repulsive in the mean and weak enough, the corresponding Schr\"odinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the $\,\delta\,$ coupling constant may be interpreted in terms of a family of squeezed potentials.Comment: LaTeX file, 7 pages, no figure

A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.Comment: 19 pages, one figure; introduction amended and some references added, to appear in CM

Band spectra of rectangular graph superlattices

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-S" type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling.Comment: 16 pages, LaTe

Quantum ergodicity for graphs related to interval maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.Comment: 20 pages, 1 figur