There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

We review some recently published methods to represent atomic neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighborhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.

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On representing chemical environments

A general interpolation method for constructing smooth molecular potential energy surfaces (PES’s) from ab initio data are proposed within the framework of the reproducing kernel Hilbert space and the inverse problem theory. The general expression for an a posteriori error bound of the constructed PES is derived. It is shown that the method yields globally smooth potential energy surfaces that are continuous and possess derivatives up to second order or higher. Moreover, the method is amenable to correct symmetry properties and asymptotic behavior of the molecular system. Finally, the method is generic and can be easily extended from low dimensional problems involving two and three atoms to high dimensional problems involving four or more atoms. Basic properties of the method are illustrated by the construction of a one‐dimensional potential energy curve of the He–He van der Waals dimer using the exact quantum Monte Carlo calculations of Anderson et al. [J. Chem. Phys. 99, 345 (1993)], a two‐dimensional potential energy surface of the HeCO van der Waals molecule using recent ab initio calculations by Tao et al. [J. Chem. Phys. 101, 8680 (1994)], and a three‐dimensional potential energy surface of the H+3 molecular ion using highly accurate ab initio calculations of Rohse et al. [J. Chem. Phys. 101, 2231 (1994)]. In the first two cases the constructed potentials clearly exhibit the correct asymptotic forms, while in the last case the constructed potential energy surface is in excellent agreement with that constructed by Rohse et al. using a low order polynomial fitting procedure.

A general method for constructing multidimensional molecular potential energy surfaces from ab initio calculations

▪ Abstract This paper describes the reproducing kernel Hilbert space (RKHS) method for constructing accurate, smooth, and efficient global potential energy surface (PES) representations for polyatomic systems using high-level ab initio data. The RKHS method provides a rigorous and effective framework for smooth multivariate interpolation of arbitrarily scattered data points and also for incorporating various physical requirements onto the PESs. Smoothness, permutation symmetry, and the asymptotic properties of polyatomic systems can be incorporated into the construction of reproducing kernels to render globally accurate PESs. Tensor products of one-dimensional generalized-spline-reproducing kernels are amenable to a fast algorithm, which makes a single evaluation of RKHS PESs essentially independent of the number of interpolated ab initio data points. This efficient implementation enables the study of the detailed dynamics of polyatomic systems based on high-quality RKHS PESs.

Constructing multidimensional molecular potential energy surfaces from ab initio data.

A global, single‐valued ground‐state H2O potential surface for the reaction O(1D)+H2→OH+H has been constructed from a new set of accurate ab initio data using a general multidimensional interpolation method. The ab initio calculations are of the multireference, configuration interaction variety and were carried out using augmented polarized triple zeta basis sets. The multidimensional method is formulated within the framework of the reproducing kernel Hilbert space theory. The H2O potential is expressed as a many‐body sum of a single one‐body term, three two‐body terms, and a single three‐body term. The one‐body term is the dissociation energy to the three‐atom limit 2H(2S)+O(3P). The two‐body terms are two O–H and one H–H adiabatic diatomic potentials of lowest energy. Each diatomic term is obtained by interpolating a discrete set of ab initio data using a one‐dimensional, second‐order, distancelike reproducing kernel. The three‐body term is obtained by interpolating the difference of the H2O ab initio d...

A global H2O potential energy surface for the reaction O(1D)+H2→OH+H

Semiclassical many-mode floquet theory

In this article a global potential energy surface for the 1A′′ state of H2O based on application of the reproducing kernel Hilbert space interpolation method to high quality ab initio results is presented. The resulting 1A′′ surface is used in conjunction with a previously determined 1A′ surface to study the O(1D)+H2(HD,D2) reaction dynamics, with emphasis on the influence of the 1A′′ excited state on measurable properties such as the reactive cross sections, rate coefficients, and product state distributions. There is a reactive threshold of about 2 kcal/mol on the 1A′′ surface, and even at 5 kcal/mol, the 1A′′ reactive cross section is only a small fraction (∼20%) of the barrierless 1A′. However, the 1A′′ surface populates very specific product vibrational states (v=3–4) and gives strongly backward peaked differential cross sections, so certain types of measurements are quite sensitive to the presence of this excited state. In particular, better agreement is found with experimental vibrational and angul...

Tak-San Ho

Papers

A general method for constructing multidimensional molecular potential energy surfaces from ab initio calculations

Constructing multidimensional molecular potential energy surfaces from ab initio data.

A global H2O potential energy surface for the reaction O(1D)+H2→OH+H

Semiclassical many-mode floquet theory

A global A-state potential surface for H2O: Influence of excited states on the O(1D)+H2 reaction