Theory and computational methods for studies of nonlinear phenomena in laser spectroscopy. I. General formalism

A general formalism will be outlined, which uses steady-state wave functions for the study of nonlinear phenomena occurring in molecular systems interacting with intense electromagnetic fields. The steady-state approach has the advantage of being free from the secular divergencies and normalization terms which appear in a perturbation expansion of the time-dependent wave function. A physical interpretation of the steady states will be given by considering the interaction between the molecule and the quantized electromagnetic field. The steady states appear as states of the combined system molecule plus electromagnetic field, with eigenvalues corresponding to the energy levels of the combined system in a semiclassical approximation. Evolution operators will be introduced and used to derive formulas for n-photon transition probabilities between molecular states both in the ordinary configuration Hilbert space and in the composite Hilbert space spanned by the steadystate wave functions.     

Natural energy orbitals and the one-particle Green's function

It is shown how the properties of the one-particle Green's function lead naturally to the definition of the so-called natural energy orbitals. These orbitals allow the fully correlated total energy of a system to be written in Hartree–Fock-like fashion and might therefore provide a bridge between sophisticated correlated wave functions and approximate theories of chemical structure and reactivity based on a Hartree–Fock-like energy expression. Moreover these orbitals form the basis for a self-consistent scheme to calculate the one-particle Green's function. The relation between these natural energy orbitals and the extended Koopmans' theorem is considered. Finally it is shown that the exactness of the lowest extended Koopmans' ionization potential implies the linear independence of the corresponding Dyson orbital from all other Dyson orbitals.     

Coordinate measurements and operators

Relativistic quantum-field theory provides the machinery for calculating wave functions or probability amplitudes depending upon space-time coordinates. The currently accepted theory, however, fails to provide position operators and a means of measuring particle coordinates that are consistent with Dirac's properties of physical observables. This is because it calls for a space position probability distribution at a specified time. This paper shows, however, that space-time event coordinate operators, together with a corresponding measurement procedure, can be found that are consistent with Dirac's requirements. This is done through a reinterpretation of the amplitudes computed by field theory and does not involve any change in that mathematical formalism. The measurement of the space-time coordinates of an event is accomplished by detecting the absorption of a photon by a particle from each of two light pulses designed to overlap at a given point at a given time. If a final emitted photon has an energy whose sum with the final particle energy approximately equals the sum of the mean energies of the pulses, then the absorption of the two pulse photons must certainly have taken place within a distance the order of a Compton wavelength of the small space-time region of overlapping pulses. This is clear from the fact that the high energy required to confine the pulses to very small volumes must throw a particle absorbing them far off the mass shell. Thus the absorption of the two photons throws the particle into a narrowly confined spatial wave function that must decay extremely rapidly—to within a Compton wavelength, a delta function in space-time. This delta function is the eigenfunction of space-time coordinate operators X^μ and is the scalar product of vectors in a Hilbert space spanned by spin–space-time kets large enough to contain the operators of the Poincaré group. These event operators transform properly under the action of Poincaré operators but do not commute with the mass. If the Compton wavelength is not negligible compared to the accuracy desired in the coordinate measurements, individual coordinate measurements are no longer possible. Nevertheless, a large number of repeated coordinate measurements can be carried out to produce a coordinate probability distribution. This distribution can be unfolded to find a true coordinate probability distribution if the charge form factor is known from basic theory. An analysis of laboratory particle detection techniques shows that they actually determine space coordinates and energy rather than spatial coordinates at a given time. When this fact is included, the Klein–Nishina formula can be derived using the electromagnetic four-vector potential as the photon probability amplitude wave. To clarify the meaning of the observables, a mass-momentum measurement is described.     

Etude de la structure du cortège électronique par la méthode des états de valence

Starting with Dirac's theory, we build up a Hamiltonian for an atomic system with several electrons. The investigation of different ways of constructing the state space of the polyelectronic system leads to the definition of “electronic configuration” and “valence state”. Using these concepts a method for calculating the atomic wave functions is set forth, which allows a precise determination of spectral term energies.     

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

A new developed approximation is used to obtain the arbitrary l‐wave bound and scattering state solutions of Schrödinger equation for a particle in a hyperbolic‐type potential. For bound state, the energy eigenvalue equation and unnormalized wave functions in terms of Jacobi polynomials are achieved using the Nikiforov–Uvarov (NU) method. Besides, energy eigenvalues are calculated numerically for some states and compared with those given in the literature to check accuracy of our results. For scattering state, the wave function is found in terms of hypergeometric functions. Furthermore, scattering amplitude and phase shifts are achieved using scattering solutions. Also it is shown that the energy eigenvalue equation obtained from analytic property of scattering amplitude is same with one obtained using NU method. © 2015 Wiley Periodicals, Inc.     

Degree of electron–nuclear entanglement in molecular states

The degree of electron–nuclear entanglement in molecular states is analyzed. This entanglement has, generally, two sources: delocalization of the electronic and nuclear wave functions and vibronic coupling. For a diatomic molecular ground‐state with a single potential energy minimum, it is demonstrated that the entanglement is a function of the product of the vibrational energy and the Born–Huang potential energy correction evaluated at the minimum. In the case of a double‐well potential energy surface, the deviation from maximal entanglement is determined by the overlap of the electronic and nuclear wave functions evaluated at and around the two minima. The adiabatic states of the E⊗ϵ Jahn–Teller model are shown to be maximally entangled and a relation between the degree of entanglement and Ham's reduction factor for this model is derived. Numerical calculations in the E⊗ϵ model demonstrate a nontrivial relation between entanglement and vibronic coupling. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 526–533, 2000     

On the properties of cumulant expansions

Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which have been obtained so far by using algebraic considerations. We give general expressions for their infinitesimal and finite transformations and represent a cumulant wave operator through an integration over a path in the Hilbert space. Cases are investigated where this integration can be done exactly. An expression of the ground-state wave function in terms of the cumulant wave operator is derived. In the second part of the article, we derive the cumulant counterpart of Faddeev's equations and show its connection to the method of increments. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 377–389, 1998     

Methods of composite molecular wave functions. I. Variational principle and multiconfiguration SCF theory

The Silverstone–Stuebing variational principle for the discontinuous wave functions of one-electron systems is generalized for many-electron systems. The variational functional of energy takes real or complex value. The condition that it is real is given. Using the generalized variational principle, a multiconfiguration SCF theory for the composite molecular wave function is formulated. According to the theory, we may divide the whole space into space-filling cells, solve the SCF equations in each cell and build up the wave functions of the system by gathering the wave functions obtained in the cells. For use in the basis-set expansion method, the SCF equations are rewritten as matrix forms in which only one- and two-center integrals appear if an expansion center is located in each cell.     

Density of non-Bloch electron states in perfect cubic crystals

This paper gives an abbreviated method for the calculation of the density of states of a crystal on the basis of that band theory in which the crystal electron states are represented by the standinglike wave functions classified according to the point-group symmetry species. The crystal is a large but finite sphere filled regularly with atoms, and the wave functions are quantized at the boundary of the sphere. The Bloch theorem is not satisfied in this theory since the wave functions are not basis functions of the irreducible representations of the translation subgroup. On the other hand, a theorem is established that the density of states can be made up of contributions given by all irreducible representations of the crystal point group, any contribution being proportional to the square of the dimension of the irreducible representation. In distinction to a former approach, the band structure is calculated solely from the energy eigenvalues obtained with the aid of the diagonalization process of the Wannier–Slater differential operator. A simple cubic lattice with an s atomic orbital on each lattice site is taken as an example, and the results are compared with Bloch's theory.     

Localization measure and maximum delocalization in molecular systems

A mathematically well-defined measure of localization is presented based on Mulliken's orbital populations. It is shown that this quantity equals 1 for core- and lone-pair orbitals, 2 for two-atomic bonds, 6 for benzene rings, etc., and it is applicable for delocalized canonical HF orbitals as well. The definition of this quantity is general in the sense that ab initio MOS with overlapping AO expansion, and semiempirical wave functions using the ZDO approximation as well, can be treated. The localization quantity is essentially “intrinsic,” i.e., no subdivision of the molecule is required. For N-electron wave functions, mean delocalization can be defined. This measure is not invariant to unitary transformations of the one-electron orbitals, characterizing in this way the localized or extended representation of the N-electron wave function. It can be proven, however, that for unitary transformed wave functions a maximum delocalization exists which depends only on the physical (N-electron) properties of the molecule. It is shown that inhomogeneous charge distribution can cause strong electron localization in molecular systems. The delocalization of the canonical Hartree–Fock orbitals, the Parr–Chen circulant orbitals, and the optimum delocalized orbitals is studied by numerical calculations in extended systems.     

Localization of multiphoton ionization/dissociation resonance wave functions in AC fields

Although it has been proved before [J. Chem. Phys. 101, 9716 (1995)] that the complex scaled photoionizing/photodissociating resonances are associated with square integrable functions (only when the time-dependent Hamiltonian is represented in the velocity or acceleration gauges), it is proved here that in the length gauge a narrow resonance wave function (and not a broad one!) may diverge exponentially at some period of time but yet decays exponentially in space at some other time. When the exterior scaling method is used and the coordinate is rotated into the complex plane by an angle less than 180° the resonance quasi-energy state (in the length gauge) decays to zero, like a bound state, at any given time. The localization of the resonance quasi-energy (Floquet) states in the length gauge as the laser frequency vanishes (i.e., when the field is varied slowly and can be considered as dc field) is proven and found to be consistent with Herbst and Simon's proof [Commun. Math. Phys. 64, 279 (1978); Ibid, 80, 181 (1981)] for atoms in dc Stark fields. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 279–285, 1997     

A minimum principle for atomic systems allowing the use of discontinuous wave functions

In this paper a variational principle proposed by Hall [1] is shown to be a minimum principle for coulombic systems. Into this principle it is possible to admit a larger class of trial wave functions than is possible in the conventional variational treatment, including wave functions with discontinuities. It is further shown that the upper bounds given by this treatment are always at least as good as that given by the Rayleigh–Ritz method. The theory is then applied to the hydrogen atom and upper bounds to the energy are calculated for various “cutoff” wave functions. It is usually possible to define an optimum “cut off” distance which minimizes the upper bound.     

Generalized electronic diabatic scheme: Diagonalizing the electronic Hamiltonian for artificial molecular systems. How do molecular meccanos move?

A quantum–classic model is presented and used to describe systems ranging from normal molecules up to electronic systems sensed in real space. The quantum system is a set of n‐electrons; a positive background in real space completes the model. A generalized electronic diabatic (GED) theory is introduced. The diabatic functions diagonalize the electronic Hamiltonian for any arrangement of the positive background. Physical quantum states are represented as linear superpositions in the diabatic basis; this latter is always fixed. For systems sensed in real space, the coefficients of the linear superposition are functions of the real space configuration coordinates. Physical changes are produced by interactions with external sources/sinks of energy. An interaction couples different diabatic states; diagonalizing the electronic Hamiltonian plus the couplings leads to new coefficients describing physical states. Among other things, these couplings can be used to simulate the effects produced by scanning tunneling microscopy, atomic force, and transmission electron microscopy on substrates located in real space. The important thing is that time–evolution in electronic Hilbert space can be related to actual motion in real space. The experiment of lateral hopping of a substrate on a metallic surface induced by vibration excitation and followed with scanning tunneling microscope is discussed. A result of the present work is that motion of molecular meccanos reflects then time–evolution in electronic Hilbert space. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Error analysis of variationally optimized upper- and lower-bound wave functions for H

Wave functions which are a linear combination of H-type elliptical orbitals are optimized to provide either an upper bound or a lower bound to the H ground state. For the latter, Temple's formula is used. Three criteria are considered to determine the relative accuracy of these wave functions: (i) energy (calculated versus exact eigenvalue); (ii) average error; and (iii) local energy. Although the lower-bound optimized wave functions obtained are the most accurate available for H from approximate wave functions, they are still inferior to the corresponding upper-bound wave functions by criteria (i) and (ii). In particular, using criterion (ii), it is shown numerically that the upper-bound functions are “correct to second order,” while the lower-bound functions are almost, but not quite, “correct to second order.” Despite this, the local energy analysis, criterion (iii), reveals that the lower-bound wave functions can be more accurate than the upper-bound functions in some regions of space, and hence give more accurate values for physical properties sensitive to these regions. Examples considered are the dipole–dipole and Fermi contact interactions.     

Hartree–Fock wave functions with a modified GTO basis for atoms

We have solved the atomic Hartree–Fock equations by using the algebraic approach, expanding the single-particle radial wave function in terms of a modified Gaussian type orbitals (GTOs) basis. Several atomic properties such as Kato's cusp condition for the electron density or the correct asymptotic behavior of the electron momentum density distribution are accurately verified. Additionally the energy of the atomic ground state can be obtained by using a smaller number of basis functions than in standard GTO expansions. This study has been performed for several atoms of the first three rows. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 59–64, 1997     

Concepts of convergence in mathematical chemistry

Some concepts of convergence used in mathematical chemistry are briefly reviewed: number convergence, uniform and non-uniform function convergence, convergence in norm and binary product, operator convergence, computer convergence, etc. Some properties of the abstract Hilbert space and some of its realizations in mathematical chemistry are discussed. Finally, it is pointed out that the scattering wave functions of importance in the theory of chemical reactions are limit points of theL ²Hilbert space — not in the norm, but in the sense of a non-uniform point-by-point convergence, which is of essential value in practical applications.     

A parallel Green's operator for multidimensional quantum scattering calculations

A parallel algorithm for computing multidimensional scattering wave functions is introduced. The inhomogeneous scattering (Lippmann–Schwinger) equation is solved within the discrete variable representation with absorbing boundary conditions, using iterative (Krylov) methods. A parallel Green's operator enables one to distribute the wave function to orthogonal subspaces in which it is processed in parallel. Application to a model problem of electron scattering in a three-dimensional rectangular quantum wire is given. Speedup is demonstrated with an increasing number of processors and with increasing dimensions and/or sampling density. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 167–173, 1998     

Natural polyelectron population analysis

A method to perform a polyelectron population analysis of correlated molecular orbital wave functions on the basis of natural atomic orbitals (NAO s), as given by Weinhold, is presented. The method allows calculations of the probabilities of finding various types of electronic events occuring in some target AO positions, including the contributions of ionic and covalent resonance structures. This method is general and neither the theory nor the developed algorithm limit the number of electrons and holes that can be considered. Thus, the analyzed MO wave function can be a usual CI or a MCSCF one, and apart from Weinhold's NAO s. any other type of orthogonal AO s can be used as analyzers, provided that these AO s are linear combinations of the SCF-AO s. Numerical applications are given for ethylene, formaldehyde, butadiene, and acroleine, by adopting various AO basis-set levels (STO ?4G , 4–31G , and 6–31G ) and by analyzing correlated wave functions (CISD ). Improvements in the polyelectron populations when increasing the quality of AO basis sets and the corresponding valence NAO s are revealed by several examples. Furthermore, it is shown that the electroegativity of oxygen in acroleine only has an effect on contributions of ionic and covalent resonance structures, but not on delocalization of the double bonds. 1993 John Wiley & Sons, Inc.     

Finite temperature string method for the study of rare events

A method is presented for the study of rare events such as conformational changes arising in activated processes whose reaction coordinate is not known beforehand and for which the assumptions of transition state theory are invalid. The method samples the energy landscape adaptively and determines the isoprobability surfaces for the transition: by definition the trajectories initiated anywhere on one of these surfaces has equal probability to reach first one metastable set rather than the other. Upon weighting these surfaces by the equilibrium probability distribution, one obtains an effective transition pathway, i.e., a tube in configuration space inside which conformational changes occur with high probability, and the associated rate. The method is first validated on a simple two-dimensional example; then it is applied to a model of solid-solid transformation of a condensed system.