Van Vleck Formula Research Articles

The Synthesis, Crystal Structure, and Magnetic Properties of Mono-Scorpionate Eu(III) Complexes

Three Eu3+ complexes containing a neutral tripodal ligand possessing a predictable coordination mode have been obtained and studied. The trispyrazolylmethane complexes have an aqua ligand in the coordination sphere, forming both the mononuclear species [Eu(HCPz3)H2O(NO3)3] and the dimer [Eu(HCPz3)H2O(CF3SO3)3]2, having a Chinese lantern structure, whereas the use of the methylated tripod leads to the water-free complex, [Eu(HC(PzMe2)3)(NO3)3]. A qualitative analysis of the magnetic susceptibility of polycrystalline samples demonstrated that the magnetic properties can be described using a simple Van Vleck formula with spin–orbit coupling parameter (λ = 383 ÷ 406 cm−1) close to the values for free Eu3+ ions. The stereochemical analysis of the coordination environment of [Ln(HC(PzMe2)3)(NO3)3] has shown that these complexes can be used as diamagnetic model systems to obtain information on the crystal field effects in the paramagnetic monoradical complexes, [LnRad(NO3)3], since both types of compounds have the same type of coordination polyhedron (symmetry point group D3h) and very close Ln–donor atom distances.

The Van Vleck Formula on Ehrenfest Time Scales and Stationary Phase Asymptotics for Frequency-Dependent Phases

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on Ehrenfest time scales, i.e. \(\hbar \)-dependent time intervals which most commonly take the form \(|t| \le c|\log \hbar |\). Our derivation is based on an approximation to the integral kernel often called the Herman–Kluk approximation, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert (Rev Math Phys 22(10):1123-1145, 2010) , this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman–Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.

Bis(toluene)chromium(I) [1,2,5]Thiadiazolo[3,4-c][1,2,5]thiadiazolidyl and [1,2,5]Thiadiazolo[3,4-b]pyrazinidyl: New Heterospin (S1 = S2 = 1/2) Radical-Ion Salts

Bis(toluene)chromium(0), Cr(0)(η(6)-C7H8)2 (3), readily reduced [1,2,5]thiadiazolo[3,4-c][1,2,5]thiadiazole (1) and [1,2,5]thiadiazolo[3,4-b]pyrazine (2) in a tetrahydrofuran solvent with the formation of heterospin, S1 = S2 = ½, radical-ion salts [3](+)[1](-) (4) and [3](+)[2](-) (5) isolated in high yields. The salts 4 and 5 were characterized by single-crystal X-ray diffraction (XRD), solution and solid-state electron paramagnetic resonance, and magnetic susceptibility measurements in the temperature range 2-300 K. Despite the formal similarity of the salts, their crystal structures were very different and, in contrast to 4, in 5 anions were disordered. For the XRD structures of the salts, parameters of the Heisenberg spin Hamiltonian were calculated using the CASSCF/NEVPT2 and broken-symmetry density functional theory approaches, and the complex magnetic motifs featuring the dominance of antiferromagnetic (AF) interactions were revealed. The experimental χT temperature dependences of the salts were simulated using the Van Vleck formula and a diagonalization of the matrix of the Heisenberg spin Hamiltonian for the clusters of 12 paramagnetic species with periodic boundary conditions. According to the calculations and χT temperature dependence simulation, a simplified magnetic model can be suggested for the salt 4 with AF interactions between the anions ([1](-)···[1](-), J1 = -5.77 cm(-1)) and anions and cations ([1](-)···[3](+), J2 = -0.84 cm(-1)). The magnetic structure of the salt 5 is much more complex and can be characterized by AF interactions between the anions, [2](-)···[2](-), and by both AF and ferromagnetic (FM) interactions between the anions and cations, [2](-)···[3](+). The contribution from FM interactions to the magnetic properties of the salt 5 is in qualitative agreement with the positive value of the Weiss constant Θ (0.4 K), whereas for salt 4, the constant is negative (-7.1 K).

Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x′, E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclassical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et al. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepúlveda-Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.

Mathematical modelling and analysis of Asian options with stochastic strike price

The goal of this article is to introduce strike Asian options on stochastic average and to price them using a variable reduction technique. The problem of solving the associated Black–Scholes equation is reduced to finding the heat kernel of the operator . This is done by two approaches: using a geometrical method and then applying the van Vleck formula; using a moments method to get a double infinite series formula involving Hermite polynomials.

Off-Center Coherent-State Representation and an Application to Semiclassics

By using the overcompleteness of coherent states we find an alternative form of the unit operator for which the ket and the bra appearing under the integration sign do not refer to the same phase-space point. This defines a new quantum representation in terms of Bargmann functions, whose basic features are presented. A continuous family of secondary reproducing kernels for the Bargmann functions is obtained, showing that this quantity is not necessarily unique for representations based on overcomplete sets. We illustrate the applicability of the presented results by deriving a semiclassical expression for the Feynman propagator that generalizes the well-known van Vleck formula and seems to point a way to cope with long-standing problems in semiclassical propagation of localized states.

Proton magnetic resonance spectra ofYH3andLuH3

A paper published by some of the present authors showed an unexpected doublet structure in the rigid-lattice proton magnetic resonance spectra of $\mathrm{Y}{\mathrm{H}}_{3}$ and $\mathrm{Lu}{\mathrm{H}}_{3}$. In the present study accurate spectra which contain no doublet features are obtained for $\mathrm{Y}{\mathrm{H}}_{2.96}$ and $\mathrm{Lu}{\mathrm{H}}_{3}$ using the technique of magic echoes. It is shown that the original spectra were distorted by time shifting of the time-domain data. The applicability of using magic echoes with these two systems is justified theoretically and by comparison to free induction decays. The measured second moments of the present spectra agree well with values predicted by the Van Vleck formula, demonstrating that the broadening of the present spectra is almost entirely dipolar.

Computer calculation of the Van Vleck second moment for materials with internal rotation of spin groups

This paper describes m2rc3, a program that calculates Van Vleck second moments for solids with internal rotation of molecules, ions or their structural parts. Only rotations about C 3 axes of symmetry are allowed, but up to 15 axes of rotation per crystallographic unit cell are permitted. The program is very useful in interpreting NMR measurements in solids. Program summary Title of the program: m2rc3 Catalogue number: ADUC Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUC Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland License provisions: none Computers: Cray SV1, Cray T3E-900, PCs Installation: Poznań Supercomputing and Networking Center ( http://www.man.poznan.pl/pcss/public/main/index.html) and Faculty of Physics, A. Mickiewicz University, Poznań, Poland ( http://www.amu.edu.pl/welcome.html.en) Operating system under which program has been tested: UNICOS ver. 10.0.0.6 on Cray SV1; UNICOS/mk on Cray T3E-900; Windows98 and Windows XP on PCs. Programming language: FORTRAN 90 No. of lines in distributed program, including test data, etc.: 757 No. of bytes in distributed program, including test data, etc.: 9730 Distribution format: tar.gz Nature of physical problem: The NMR second moment reflects the strength of the nuclear magnetic dipole–dipole interaction in solids. This value can be extracted from the appropriate experiment and can be calculated on the basis of Van Vleck formula. The internal rotation of molecules or their parts averages this interaction decreasing the measured value of the NMR second moment. The analysis of the internal dynamics based on the NMR second moment measurements is as follows. The second moment is measured at different temperatures. On the other hand it is also calculated for different models and frequencies of this motion. Comparison of experimental and calculated values permits the building of the most probable model of internal dynamics in the studied material. The program described in this paper calculates the second moment for solids with rotation of different groups of spins with C 3 symmetry. Method of solution: The rotation of molecules or their parts, for example CH 3 groups, is simulated as a random walk process by rotating each individual group of spins about its symmetry axis by an angle allowed by the type of symmetry. It is not a continuous rotation, but is in the form of jumps between consecutive positions allowed by the symmetry of the rotating group. Such a model of rotation fulfills assumptions on which theoretical equations used in NMR are derived. The value of Van Vleck's second moment averaged by this rotation is evaluated. The degree of averaging depends on the number of rotational jumps simulated during calculation. This number is then expressed in terms of the frequency of rotation and finally into the temperature. As a result we obtain simulated values of the NMR second moment as a function of temperature. Restrictions on the complexity of the problem: The only restriction is the number of spins for which calculations can be performed in a reasonable amount of CPU time. This restriction is therefore a combination of the number of spins in the unit cell, number of unit cells included in the calculation, and the speed of the computer used. The tested version of the program was compiled for a maximum number of 6250 spins, arranged in 125 unit cells. There are 15 axes of rotation allowed per unit cell. Any of these restrictions can be overcome by increasing the dimensions of the appropriate arrays in the program. The dimensions given in the program are sufficient for analysis of most of the NMR data which one can find in the scientific literature. This is due to the fact that the magnetic dipole–dipole interaction decreases with the third power of distance between spins, and calculations including spins up to a distance of about 2.0 nm give a final accuracy of the second moment equal to about 1%, while experimental values are determined with 5% accuracy or even worse. The program was designed to handle any combination of complex rotations, but only about C 3 axes. Overcoming this restriction by introducing the possibility of C 6 or C 4 rotations would require some changes in the program. They may be quite easily introduced by an experienced programmer.

Quantum mechanics of a free particle beyond differential equations

With Feynman's path- integral method we can obtain the quantum mechanics of a quantum system like a free particle outside Schroedinger's method of differential equations and Heisenberg's method of algebra. The work involves obtaining the quantum propagator Kf, of the system which leads to summation over infinite number of paths. With Van Vleck's formula in one dimension, the classical propagator Kcl for a free particle is computed as the analytical result. This then serves as a yardstick for justifying the theoretical method used to compute the quantum propagator, kf, by direct path summation. The graphical display of the results shows that the Feynman – Schulman's checkerboard model used to enumerate the paths is reliable. Furthermore, this work shows that by windowing off a large number of paths and weighting the rest non uniformly we can compute the required propagator. The weights used in this case are the random and exponential window functions, wr and we respectively, which yield kwr and kwe to compare with kf. Key words: Propagator, action, path - integral, window function, free particle. (Global Journal of Pure and Applied Sciences: 2003 9(4): 561-566)

Paramagnetic susceptibility simulations from crystal field effects on Nd3+ in AgNd(WO4)2

Polycrystalline AgNd(WO4)2 has been prepared by a in solid-state reaction. X-ray diffraction measurements indicate that this compound crystallizes with a scheelite-type tetragonal structure. Absorption and emission spectra have been measured, using the technique of the crystal field parameters, to obtain the energy levels scheme for Nd3+ ions in the crystalline matrix. These data indicate the existence of a structural local disorder associated with a random distribution of the ions of Ag+ and Nd3+. The temperature variation of the paramagnetic susceptibility in the range from 2 to 300 K has been studied. The Van Vleck formula has been used to calculate the temperature variation of the paramagnetic susceptibility and it was compared with the experimental results. Particularly at low temperatures, satisfactory agreement has been obtained.

Semiclassical quantization rule for the bound-state spectrum in quantum dots: Scattering phase approximation

We study the quantum propagator in the semiclassical limit with sharp confining potentials. Including the energy-dependent scattering phase due to sharp confining potential, the modified Van Vleck formula is derived. We also discuss the close relations among quantum statistics, discrete gauge symmetry, and hard-wall constraints. Most of all, we formulate a quantization rule that applies to both smooth and sharp boundary potentials. It provides an easy way to compute quantized energies in the semiclassical limit and is extremely useful for many physical systems.

Quadratic Wiener functionals and dynamics on Grassmannians

We introduce a new framework to study the structure of the probabilistic laws of quadratic Wiener functionals in terms of dynamics on Grassmannians. The key point is to identify the Jacobi equation derived by an analogue of Van Vleck formula with the space of its solutions according to M. Sato's fundamental idea.

Optical spectroscopy and crystal-field effects on the paramagnetic susceptibility of rare-earth germanates GaRGe 2O 7, R=Pr, Nd

Optical absorption and photoluminescence measurements at 9 K were performed on stoichiometric GaRGe 2O 7 and on R-doped LaGaGe 2O 7, R=Pr 3+, Nd 3+, polycrystalline samples. In this monoclinic matrix, space group P2 1/c (no. 14), the lanthanide ion occupies a single crystallographic point site with symmetry C 1. From the crystal-field analysis of the optical spectra, energy level schemes and an expression of the associated wavefunctions for the 4f 2 and 4f 3 configurations were derived. In both cases, the fitting of experimental Stark level energies and the phenomenological calculation of free-ion as well as crystal-field parameters (CFPs) were performed for the approximate C s (C 2) symmetry. Despite the low symmetry, very satisfactory correlations between experimental and simulated energy level schemes were obtained, with root mean square deviations σ=10.1 and 14.8 cm −1 for Pr 3+ and Nd 3+, respectively. Energy levels and the composition of their wavefunctions were checked for both configurations through a calculation of the thermal evolution of the paramagnetic susceptibility χ, according to the Van Vleck formula. The same calculation was accomplished with CFPs resulting from the semi-empirical simple overlap model (SOM), considering the real point symmetry of R and C 1. Over the whole measured range, 1.7–300 K, a very good reproduction of the observed χ vs. T is yielded by each of the phenomenological and SOM calculated collections of CFPs.

Paramagnetic susceptibility simulations from crystal field effects on Nd3+ in magnesium borate MgNd(BO2)5

Abstract From the crystal field analysis of optical absorption spectra on monoclinic S.G. P21/c MgNd(BO2)5, where Nd3+ occupies a single crystallographic position with no symmetry elements, energy level schemes and an expression of the associated wavefunctions for the 4f3 configuration of Nd3+ have been derived, considering approximate C2v and Cs (C2) symmetries. Despite of the low symmetry of Nd3+, resulting rms deviations between calculated and experimental levels are very satisfactory. The composition of the crystal field wavefunctions from both sets of phenomenological free-ion and crystal field parameters has been checked through a calculation of the thermal evolution of the paramagnetic susceptibility χ, according to the Van Vleck formula. The same simulation was performed with crystal field parameters (CFPs) resulting from the ab initio simple overlap model (SOM), considering the Nd3+ real point symmetry, C1. Over the whole measured range, 1.7–375 K, and especially at low temperatures, a very good reproduction of the experimental χ vs. T is yielded by each of the experimental and SOM calculated collections of CFPs. The oversimplified SOM model seems thus to be useful providing reasonable estimations of crystal field strength parameters.

Calculation of the NMR second moment for materials with different types of internal rotation

The measurements of the NMR second moment can reveal information about structural and dynamical details of the sample providing that analysis of the experimental results on the basis of the Van Vleck's formula can be performed. In this paper the method of calculating the Van Vleck's second moment for solids with complex internal motions is presented. The method is based on simulating any desired motion of atoms within a block of unit cells large enough to reflect the property of the macroscopic sample and calculating the second moment value averaged by this motion. The detailed description of this method is given for the case of rotation of molecules or groups of atoms. The algorithm of the computer program which performs calculation based on the described method is also presented. Examples of application of the described method are listed.

Paramagnetic susceptibilities of trivalent neodymium in LiYF4

The paramagnetic susceptibilities of the 1.5%-doped LiYF4:Nd3+ single crystal were re-recorded using a SQUID in the temperature range from 1.7 K to 300 K. The anisotropic values χ‖ and χ⊥ are simulated by means of optimised crystal field wave functions inserted in the Van Vleck formula. The agreement between the simulated and experimental data is satisfying, especially in the low temperature range.

Crystal Field and Exchange Interactions in ThMn12-Type RZn12 Compounds (R = Tb, Tm)

Susceptibility measurements on TbZn 12 , TmZn 12 and YZn 12 have been utilised to establish the crystal field and exchange interactions in these compounds. TbZn 12 and TmZn 12 order antiferromagnetically at low temperatures (with Neel temperatures TN of 13 and 4 K, respectively) and then they are paramagnetic above T N YZn 12 has been found to be a Pauli-type paramagnet. The paramagnetic susceptibility was calculated using the van Vleck formula and taking into account the s-f polarisation effects and interionic RKKY-type exchange interactions. The leading crystal field parameters B 2 0 and B 4 0 have been obtained to be 13.5 mK and -0.037 mK for TbZn 12 and -14.2 mK and 0.036 mK for TmZn 12 , respectively.

Complex trajectory method in semiclassical propagation of wave packets

We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.

Complete set of magnetic parameters in a novel nickel(II) chelate

A theoretical analysis of powder magnetic data based on the diagonalization of the spin Hamiltonian matrices is outlined. The first and second derivatives of the eigenvalues are determined from the fourth-order polynomial fit; they enter the van Vleck formula for the paramagnetic susceptibility. The procedure is capable of subtracting the complete set of magnetic parameters from powder susceptibility data. A five-member Ni(II) metallocycle ( S=1) was investigated by SQUID measurements down to 2 K and a substantial drop of the χT curve below 25 K was found. A reconstruction of the components of the magnetic susceptibility was done. An independent fitting procedure using four fundamental magnetic parameters gave Λ xx =42.8±0.1, Λ yy =49.9±0.1, Λ zz =26.6±0.1 au and zJ/ hc=−2.06±0.01 cm −1. These data reproduce the enlarged set of magnetic parameters well so that a selfconsistency of magnetic parameters is proven.

Improvement on Van Vleck's formula for diatomic nonadiabatic energy shifts

A formula ΔE ivJ ≈ 〈 v J |[2 m( E ivJ − U J i ) B 2 ij + (d B ij /d R) 2]/ ΔU ij | v J 〉/ m 2, where m is the reduced molecular mass, B ij = 〈 i| ∂/ ∂R| j〉 is the radial nonadiabatic coupling matrix element for two interactive electronic states i and j, and ΔU ij = U J i ( R) − U J j ( R) is the difference in the corresponding effective adiabatic internuclear potentials, is offered to approximate a nonadiabatic shift of the rovibronic term E ivJ arising from regular homogeneous perturbations in diatomic molecules. The formula has been numerically tested for B 1Σ + u , B′ 1Σ + u , C 1Π u and D 1Π u states of molecular hydrogen. It is demonstrated that the present approximation is more accurate than the original Van Vleck formula for high v-values and has a high efficiency for nondiagonal systems with significant continuum contribution.