Generalized Brillouin theorem and generalized forces in SCF and MC SCF theories and their Hartree-Fock limits

A formula for the change of energy is derived, valid for any infinitesimal change of the hamiltonian and the wave function in CI, SCF and MC SCF theories. A general form of MC Hartree-Fock equations is derived in a somewhat new form. A closed MC theory is defined and a condition of self-consistency is derived for this theory. The role of deorthogonalization of spin orbitals is discussed. The Hellmann-Feynman theorem and the generalized Brillouin theorem are rederived in a unique and simple way. It is shown that in the case of the closed MC SCF theory the Brillouin theorem derived by Levy and Berthier can be strengthened.

Formulae for generalized forces are derived, valid for closed-shell, openshell and multiconfigurational SCF theories also far from the Hartree-Fock limit.     

Attraction between nanoparticles induced by end-grafted homopolymers in good solvent

The interactions between nanoparticles coated by end-grafted homopolymers of comparable size are studied using Edwards' self-consistent field (SCF) theory. The equilibrium monomer density distribution is obtained by solving the SCF equations in bispherical coordinates. It is shown that, because of the spherical geometry and redistribution of monomer density, end-grafted homopolymers in good solvent environment can induce attractive interactions. This result suggests that steric stabilization of nanoparticles requires new working principles and that the equilibrium distance between nanoparticles may be tuned by end-grafted polymers.     

Off-diagonal constrained variations in open-shell SCF theory

The formalism for the off-diagonal constrained variation method is developed for open-shell SCF theory. New self-consistent pseudoeigenvalue equations are derived. The imposed constraints can be either theoretical, based on the off-diagonal hypervirial theorems, or semi-empirical, obtained by requiring that calculated transition dipoles should agree with experimental values.     

Effect of multiple reemission on the shape of the magnetic-depolarization contour

It is shown that in order to describe the level-intersection effect, taking into account repeated emission, it is necessary to solve the boundary problem. The passage of resonant polarized radiation through a plane-parallel layer under a magnetic field is considered. The method of invariant merging [8] is used to construct equations for the reflection and transmission functions. Reciprocity principles are established.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 25–30, December, 1976.     

Renormalization of QED in an external field

The Schwinger equations of QED are rewritten in three different ways as integral equations involving functional derivatives, which are called weak field, strong field, and SCF quantum electrodynamics. The perturbative solutions of these equations are given in terms of appropriate Feynman diagrams. The Green function that is used as an electron propagator in each case is discussed in detail. The general renormalization rules for each of the three equations are provided both in a non perturbative way (Dyson relations) and for Feynman diagrams.     

采用解析法对Nd:YAG单晶光纤的温度场研究

采用解析法对Nd:YAG单晶光纤激光器热效应相关的光纤温度场分布进行研究。建立了Nd:YAG单晶光纤热模型,在单晶光纤所满足的边界条件下通过解析求解热传导方程,得到在高功率808 nm泵浦光抽运下产生946 nm激光的单晶光纤温度场分布,并与传统Nd∶YAG激光晶体的温度场进行比较,然后分别与同泵浦条件下的有限元数值方法的分析结果进行研究对比,最后分析泵浦光参数、单晶光纤参数等对温度场的影响。结果表明,功率为86 W的泵浦光入射至单晶光纤端面的最高温升仅为30.98℃,明显优于同泵浦条件下传统Nd∶YAG晶体的端面温升结果94.37℃,与有限元数值法得到的Nd:YAG单晶光纤19℃温升结果相比,该解析法结果更接近于实验的测量值31℃,能够更精确描述晶体光纤温度场。本文可对单晶光纤激光器热效应的精确研究提供一定参考,进而有利于提高单晶光纤激光器的性能。     

Theory of the crystal-liquid phase transition

The SCF approximation is used in a statistical theory of melting based on equality of the chemical potentials. The results for the melting points of the inert gases are close to the observed values.I am indebted to Academician N. N. Bogolyubov for a discussion, and to V. B. Glasko and E. Kh. Kats for performing the numerical calculations on the system of transcendental equations.     

A class of algebraically special perfect fluid space-times

Solutions of the Einstein field equations are considered subject to the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the Weyl tensor is algebraically special, (3) the corresponding repeated principal null congruence is geodesic and shearfree. If in addition, the repeated principal null congruence is non-expanding, it follows that the twist of this congruence must be non-zero (for a physically reasonable fluid). The general line element subject to this additional restriction is derived. Furthermore, it is shown that all solutions of the Einstein field equations which satisfy (1) and exhibit local rotational symmetry, necessarily satisfy (2) and (3).This work was supported in part by the National Research Council of Canada.     

On the improvement of expectation values in the uncoupled SCF and HF perturbation theories

The exact derivation of the formula for the first-order ‘correlation’ correction to the second-order uncoupled Hartree-Fock energies, computed with approximate Hartree-Fock orbitals, is presented. A comparison with the corresponding formula derived under the assumption of the exact solution of the Hartree-Fock equations indicates that also the second-order perturbed orbitals do contribute. However, their contribution can be expressed in terms of the first-order correction to the SCF orbitals which accounts for the difference between the Hartree-Fock and SCF solution. These additional terms can be either positive or negative and can influence significantly the final result for the correlation-corrected second-order energies.     

Analytical gradients of a state average MCSCF state and a state average diagnostic

An efficient method for calculating the Lagrange multipliers and the analytical gradients of one state included in a state average MCSCF wave function is presented. It is demonstrated that the state average energy of an ‘equal-weight’ scheme is invariant to rotations within the state average subspace and that the corresponding rotations should be eliminated from the Lagrangian equations. Finally, a diagnostic is presented, which gauges the energy difference between a state defined by a state average calculation and the corresponding fully variational multi-configurational SCF state.     

On the temperature-dependent shift of the F-center levels

The temperature-dependent shift of the mean electron level is deduced employing adiabatic approximation and Hartree approximation of SCF for the F-center electron together with Einstein model for ionic motion. At high temperatures the shift rises linearly and its sign depends on the detailed structure of centre in the given state. As a consequence of different shifts for the ground and excited states, the absorption and emission band maxima are shown to shift with temperature as well. In the case of thermal ionization, the corresponding shift can cause a great enhancement of the effective frequency factor, if the current glow curve method is used.     

Analytical energy gradients for dynamical simulations of trans polyacetylene chains within the Pariser-Parr-Pople Hamiltonian

Exact analytical energy gradients for the Pariser-Parr-Pople (PPP) Hamiltonian are derived. A comparison of computer times for dynamical simulations of trans-Polyacetylene (t-PA) using analytical and numerical gradients is given. The numerical method is shown to lead to serious difficulties both computationally and from the point of view of numerical accuracy. In fact, using the analytical method, it turned out that the computational effort for the gradient calculation is negligible compared to that for the SCF iteration in each time step of a simulation. On the other hand, using the numerical method the gradient calculation is the time consuming bottle-neck of a simulation. A previously presented method for the gradient calculation in the Hückel type Su-Shrieffer-Heeger (SSH) Hamiltonian which was thought to be approximative, is shown to be exact.     

The hilbert expansion to the Boltzmann equation for steady flow

The Hilbert expansion to the Boltzmann equation is carried out for steady flow. It is shown that the first term in the Hubert series for the distribution function is a local Maxwellian leading to the steady Euler equations. The steady field equations that follow from the solution of the second term in the series are derived. The formulas for thermal conductivity and for viscosity of Hilbert that appear in the steady field equations of the second approximation are shown to be precisely the same as those obtained by Chapman and Enskog. The procedure to obtain higher approximations by Hubert's method is summarized.     

Transformation properties of the integrable evolution equations

Group-theoretical properties of partial differential equations integrable by the inverse scattering transform method are discussed. It is shown that nonlinear transformations typical to integrable equations (symmetry groups, Bäcklund-transformations) and these equations themselves are contained in a certain universal nonlinear transformation group.     

Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies

A method for analyzing multidegree-of-freedom systems having a repeated natural frequency subjected to a parametric excitation is presented. Attention is given to the ordering of the various terms (linear and non-linear) in the governing equations. The analysis is based on the method of multiple scales. As a numerical example involving a parametric resonance, panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis. Some of the analytical results are verified by a numerical integration of the governing equations.     

A new mixed self-consistent field procedure

A new approach for the calculation of three-centre electronic repulsion integrals (ERIs) is developed, implemented and benchmarked in the framework of auxiliary density functional theory (ADFT). The so-called mixed self-consistent field (mixed SCF) divides the computationally costly ERIs in two sets: far-field and near-field. Far-field ERIs are calculated using the newly developed double asymptotic expansion as in the direct SCF scheme. Near-field ERIs are calculated only once prior to the SCF procedure and stored in memory, as in the conventional SCF scheme. Hence the name, mixed SCF. The implementation is particularly powerful when used in parallel architectures, since all RAM available are used for near-field ERI storage. In addition, the efficient distribution algorithm performs minimal intercommunication operations between processors, avoiding a potential bottleneck. One-, two- and three-dimensional systems are used for benchmarking, showing substantial time reduction in the ERI calculation for all of them. A Born–Oppenheimer molecular dynamics calculation for the Na⁺₅₅ cluster is also shown in order to demonstrate the speed-up for small systems achievable with the mixed SCF.     

Simple functional-differential equations for the bound-state wave-function components

We present a new method of a direct derivation of differential equations for the wave-function components of identical-pariticles systems. The method generates in a simple manner all the possible variants of these equations. In some cases they are the differential equations of Faddeev of Yakubovskii. It is shown that the case of the bound states allows to formulate very simple equations for the components which are equivalent to the Schrödinger equation for the complete wave function. The components with a minimal antisymmetry are defined and the corresponding equations are derived.     

The finite element method for the coupled Schrödinger-KdV equations

In this Letter, we employ finite element method to study a periodic initial value problem for the coupled Schrödinger-KdV equations. For the case of one dimension, this problem is reduced to a system of ordinary differential equations by using a semi-discrete scheme. The conservation properties of this scheme, the existence and uniqueness of the discrete solutions, and error estimates are presented. In numerical experiments, the resulting system of ordinary differential equations are solved by Runge-Kutta method at each time level. The superior accuracy of this scheme is shown by comparing the numerical solutions with the exact solutions.     

Nd：YAG单晶光纤光学特性研究

本文测量了用激光加热基座法拉制的Nd:YAG单晶光纤(具有块状晶体)的吸收谱线,荧光谱线等光谱学特性.根据测得的散射特性分析了用该法拉制的单晶光纤的生长缺陷和直径波动及其与单晶光纤质量的关系.