The Consolidation of Matrix Mechanics: Born–Jordan, Dirac and the Three-Man-Paper

The Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

It is shown that the Thomas–Reiche–Kuhn sum rule, associated with the photoabsorption cross section from quantum systems, appears to be violated in the case of the quantized rigid rotator. The origin of the apparent violation is investigated, and its resolution is presented on the basis of a related system, i.e., a particle in a spherical δ-function potential whose energy spectrum approaches that of the rigid rotator when the strength of the potential becomes large.

Optical oscillator strength distribution of amino acids from 3 to 250 eV and examination of the Thomas–Reiche–Kuhn sum rule

An important challenge for lowering the cost of “solar energy” is minimising the required usage of the “active solar absorber material.” Development of ultra-thin solar cells is of paramount importance in ultimate cost reduction of solar cells—without compromising if not increasing, the efficiency of solar cells. Research in ultra-thin solar cells is fast-gaining ground. It was pointed out that basing on Thomas–Reiche–Kuhn sum rule, the amount of material required to achieve maximum optical absorption due to incident light (in the spectral region of interest for solar cells) may well be around 10 nm thick. However, it is important to devise appropriate light manipulation mechanism in conjunction with the semiconductor absorber layer of the solar cells. Plasmonics—the science and technology of confining the electric field energy in low-dimensional systems—is an important route to successfully achieve the “ultra-thin solar cells”. Plasmonics offer two routes for light manipulation—near field and far field. These mechanisms in turn enable more secondary mechanisms such as hot-carrier generation, photon up-conversion, nonlinear effects, etc. When employed optimally, these mechanisms will aid one another producing the amplifying effect on the optical absorption in the active semiconductor absorptive layer of solar cell and consequently on the efficiency of the cell. Availability of methodology and techniques for easily and cost-effectively incorporating plasmonic structure in the immediate vicinity of the semiconductor absorber of the solar cell is one of the limiting factors in achieving the ultra-thin solar cells. Plasmonic metasurfaces—2D analog of plasmonic metamaterials—are found to possess broadband optical properties required for solar cells. There is a growing body of research to implement plasmonic light trapping effects on various inorganic and organic ultra-thin solar cells. We will discuss various plasmonic light trapping mechanisms with respect to solar cells and possible directions to successful implementation in various types of industrially important solar cells.In this chapter, we will describe the following aspects: (1) concept behind plasmonic photon management, innovative plasmonic structures to confine, scatter light and modify electric field; 2D multilayers, core–shell structures, custom-designed textures and topography; (2) characterisation methods to probe the plasmonic effects, diagnosis tools employing plasmonic effects; (3) solar cell structures, adapting fabrication process of the first-generation and second-generation solar cells in the market, to make effective use of plasmonic light trapping through both far field and near effects, absorber material consideration, assessment of optical gain compared to plasmonic loss and evolution of electronic/structural defects and shunt paths; (4) challenge of upscaling and industrial plasmonic PV fabrication tool; (5) other practical/potential applications: LED, water splitting, third- and future generation solar cells, special emphasis on up-conversion; and (6) industrial viability of the plasmonic-based devices, compared to existing scenario.This chapter will explain how dimension on optoelectronic devices can be substantially thinned down by increasing light trapping efficiency through plasmonic effects.

We report the results of pump–probe optical Kerr effect (OKE) experiments performed on neat solutions of carbon tetrachloride, nitrobenzene, methyl methacrylate monomer, binary solutions of the squaraine dye indole squarylium, and the phthalocyanine dye silicon phthalocyanine-monomethacrylate, respectively, in carbon tetrachloride, and solid solutions of indole squarylium and phthalocyanine-monomethacrylate in poly(methyl methacrylate). Dispersion measurements of the dye solutions were performed in the visible one-photon resonant region of the dyes defined by their linear-absorption spectra. The dyes’ third-order molecular susceptibility response γxxxx(-ω2;ω1,-ω1, ω2) in this spectral region is markedly different, with R{γISQ}>0 and R{γSiPc}<0. Analysis of the dyes’ OKE response requires the inclusion of high-lying two-photon states and suggests that a purely electronic mechanism dominates their OKE response. The results are used to calculate the dyes’ off-resonant third-order molecular susceptibilities, which are well within the limits predicted by the Thomas–Reiche–Kuhn sum rule [ M. G. Kuzyk , Opt. Lett.25, 1183–1185 (2000)].

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the ethylene molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of this molecule. The constraints are furnished by molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for ethylene. Pseudo-DOSDs for this molecule, and for propene and 1–butene, which are based on an earlier constrained DOSD analysis for these molecules, are developed. They are used to obtain reliable results for the isotropic dipole–dipole dispersion-energy coefficients C6, for the interactions of the alkenes with each other and with 47 other species, and the triple-dipole dispersion-energy coefficients C9 for interactions involving any triple of molecules taken from ethylene, propene, and 1–butene.Key words: alkenes, dipole properties, pseudo-states, dipole–dipole and triple-dipole dispersion energies, long-range additive, non-additive interaction energies.

The connection between the nuclear electric shielding and the atomic polar tensors are shown. The electric shielding tensors are related to the polarizability and the magnetizability, and satisfy a constraint condition for the electrostatic equilibrium which is the mixed length-acceleration Thomas–Reiche–Kuhn sum rule. In addition, they can be successfully used to rationalize experimental IR intensity data, which is verified by extended basis set calculations on the water molecule.

Dipole oscillator strength distributions have been constructed and used to evaluate integrated oscillator strengths, and a variety of dipole oscillator strength properties, for ground state SO2, CS2, and OCS. Each distribution has been constructed by using experimental and theoretical photoabsorption cross sections and by subjecting the resulting dipole oscillator strength data to constraints provided by the Thomas–Reiche–Kuhn sum rule and molar refractivity data for the relevant dilute gases. The discussion includes graphical presentations of how various spectral regions of the dipole oscillator strength distributions contribute to the more important dipole properties.

The Thomas–Reiche–Kuhn sum rule, within the acceleration gauge for the transition dipole moment, is used to partition the total number of electrons in a molecule and to define atomic populations, which can be related to corresponding experimental estimates of atomic polar tensors from IR intensities. As a bond dipole moment can only be defined for diagonal atomic polar tensors, it is shown that the assumption of a C–H bond moment, transferable from molecule to molecule in the alkane series, is physically unreliable. From experimental IR intensities of methane we infer that, for the equilibrium geometry, μC–H =0.339 D, directed C−H+. Accordingly, it is argued that the theoretical bond dipole moments, estimated for the C–H bond in methane via localization procedures of SCF wave functions, are questionable, as they predict the opposite polarity. Finally, a resolution of the electric dipole moment into atomic contributions is suggested.

Isotropic dipole oscillator strength distributions (DOSDs) have been constructed for the dimethyl, diethyl and methyl–propyl ether molecules through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSDs are used to obtain recommended values for a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecules. Pseudo-DOSDs for the ethers are also constructed and used to obtain reliable results for the isotropic dipole–dipole dispersion energy coefficients for all two-body interactions of the ethers with each other and with fifty other species. In addition reliable results are also obtained for the triple–dipole dispersion energy coefficients for all three-body interactions involving the ethers. 1Dedicated to Anthony Stone, an excellent scientist and friend, on the occasion of his 70th birthday.

The photoionization cross sections of CO2 leading to the first four electronic states of CO+2 have been computed including the effects of interchannel coupling. The results were obtained in the Tamm–Dancoff approximation using the Padé-approximant C̃-functional method to solve the resulting scattering equations. All of the required matrix elements have been computed using single-center expansions and numerical integration of the resulting radial functions. An alternative approach for computing products of single-center-expanded functions is presented where the functions are transformed into a coordinate representation, then the product is computed, and finally the product is transformed back into the angular momentum representation. The computational effort required in this approach depends on the second power of the number of partial waves in contrast to the third power dependence found in methods used previously. The photoionization cross sections are obtained in the mixed dipole representation which ensures that the Thomas–Reiche–Kuhn sum rule is satisfied. In the coupled-channel approximation, the shape resonance in the 4σg→kσu channel is found to remain at the same energy and have the same width as was found in earlier single-channel calculations. Both the total cross section in the (4σg)−1 channel and the photoelectron asymmetry parameter are somewhat less affected by the resonance than in the single-channel approximation, but there is still a substantial disagreement with experimental data. The kσu shape resonance is found to modify the cross sections and asymmetry parameters in the other channels, with the largest effect being in the (3σu)−1 ionization channel. The full coupled-channel results, which include coupling among the 1πg→kπu,1πu→kπg,3σu→kσg, and 4σg→kσu channels, are found to significantly modify the cross section in the 1πg→kπu channel leading to good agreement between theory and experiment for the total ionization cross section in the (1πg)−1 channel. However, this coupling is found to significantly perturb the other channels, and in the case of the asymmetry parameters in the (3σu)−1 channel, this leads to relatively poor agreement with experimental data.

A B-spline Galerkin method for the Dirac equation

The linear response calculations in the multiconfiguration time-dependent Hartree–Fock (MCTDHF) approximation with a closed-shell-type MCSCF state as the time-independent reference state are discussed. The application to the LiH molecule with a small basis set ([4s2p1d/2s1p]) shows validity of our MCTDHF approach to the singlet ground state. Our MCSCF correlation energy is 97% of the total (=full CI) correlation energy and the MCTDHF excitation energies are in good agreements with the Δ full CI excitation energies. The Born–Oppenheimer potential energy curves for the lowest three singlet states of LiH and the corresponding vibrational level spacings, the transition moments, the oscillator strengths, and the frequency-dependent dipole polarizabilities are reported. All of these results imply the potentiality of our MCTDHF method for the future work with the larger basis set. One of such basis sets ([9s8p4d/8s7p1d]) is referentially used only at the single-configuration TDHF level, and the resultant near-Hartree–Fock polarizability and Thomas–Reiche–Kuhn sum rule is very promising.

Dipole oscillator strength distributions have been constructed, and used to evaluate integrated dipole oscillator strengths and a variety of dipole oscillator strength properties, for ground state Ne, Ar, Kr, Xe, HF, HCl, and HBr. Each distribution has been constructed by using experimental and theoretical photoabsorption cross sections and by subjecting the resulting dipole oscillator strength data to constraints provided by the Thomas–Reiche–Kuhn sum rule and molar refractivity or related data for the relevant dilute gases. The resulting dipole properties are generally the most or only reliable values available for the hydrogen halides, for the rare gases they largely augment reliable results for most of the properties already available in the literature. The results for HCl are used to discuss an interesting molecular effect in the photoabsorption spectrum of this molecule betwen ~200 eV and ~205 eV.

Newly available and highly accurate oscillator strength data, extending over the continuous energy range from the first excitation threshold to 200 eV, are used together with mixture rule estimates and other photoabsorption data to construct a refined dipole oscillator strength distribution to infinite photon energy using Thomas–Reiche–Kuhn sum rule and molar refractivity constraints. This constrained dipole oscillator strength distribution has been used to calculate a wide range of related dipole properties (Sk, k = 2, 1, 0, −1/2, −1, −3/2, −2, −5/2, −3, −4, −5, −6, −8, −10, −12; Lk and Ik, k = 2, 1, 0, −1, −2). The theoretical analysis and associated consistency checks support the high accuracy of the newly available absolute oscillator strengths for the photoabsorption of NH3.