(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

The polarized optical spectra of the ions Ti3+, V3+, Cr3+, Mn3+, Co3+, and Ni3+ in corundum single crystals have been studied at temperatures from 4.2° to 1200°K. A theory of the band strength based on the point‐charge model and p‐d mixing has been developed and applied to the data with results in fair agreement with experiment. The effects of temperature show that the vibrational‐electronic contribution to band strength is quite small at low temperature but may be appreciable at high temperatures. The crystal‐field parameters have been calculated as convergent lattice sums. The observed trigonal‐field parameter has the opposite sign from that calculated by the point‐charge model if the impurity ion is assumed to occupy an Al3+ ion position in the lattice, but has the same sign as calculated for an ion 0.1 A displaced along the c3 axis toward the empty octahedral site. Details of the spectra have been interpreted as showing that the surroundings of an ion are distorted in some electronic states.

Optical spectra of transition-metal ions in corundum

An experimental wavefunction is one that has an assumed form and that is also fitted to experimental measurements according to some well defined procedure. In this paper, the concept of extracting wavefunctions from experimental data is critically examined and past efforts are reviewed. In particular, the importance of scattering experiments for wavefunction fitting schemes is highlighted in relation to the more familiar model, the Hamiltonian paradigm. A general and systematically improvable method for fitting a wavefunction to experimental data is proposed. In this method, the parameters in a model wavefunction are determined according to the variational theorem but subject to an imposed constraint that an agreement statistic between the calculated and observed experimental data has a certain acceptable value. Advantages of the method include the fact that any amount of experimental data can be used in the fitting procedure irrespective of the number of parameters in the model wavefunction, the fact that a unique answer is obtained for a given choice of the model wavefunction, and the fact that the method can be used to model different experiments simultaneously. The wavefunction fitting method is illustrated by developing the theory for extracting a single-determinant wavefunction for a fragment of a molecular crystal, using data obtained from elastic X-ray scattering data. Effects due to thermal motion of the nuclei, secondary extinction of the X-ray scattering and different choices for the crystal fragment are treated.

Wavefunctions derived from experiment. I. Motivation and theory.

There are two schools of thought in chemistry: one derived from the valence bond and molecular orbital models of bonding, the other appealing directly to the measurable electron density and the quantum mechanical theorems that determine its behavior, an approach embodied in the quantum theory of atoms in molecules, QTAIM. No one questions the validity of the former approach, and indeed molecular orbital models and QTAIM play complementary roles, the models finding expression in the principles of physics. However, some orbital proponents step beyond the models to impose their personal stamp on their use in interpretive chemistry, by denying the possible existence of a physical basis for the concepts of chemistry. This places them at odds with QTAIM, whose very existence stems from the discovery in the observable topology of the electron density, the definitions of atoms, of the bonding between atoms and hence of molecular structure. Relating these concepts to the electron density provides the necessary link for their ultimate quantum definition. This paper explores in depth the possible causes of the difficulties some have in accepting the quantum basis of structure beginning with the arguments associated with the acceptance of a "bond path" as a criterion for bonding. This identification is based on the finding that all classical structures may be mapped onto molecular graphs consisting of bond paths linking neighboring atoms, a mapping that has no known exceptions and one that is further bolstered by the finding that there are no examples of "missing bond paths". Difficulties arise when the quantum concept is applied to systems that are not amenable to the classical models of bonding. Thus one is faced with the recurring dilemma of science, of having to escape the constraints of a model that requires a change in the existing paradigm, a process that has been in operation since the discovery of new and novel structures necessitated the extension of the Lewis model and the octet rule. The paper reviews all facets of bonding beginning with the work of Pauling and Slater in their accounting for crystal structures, taking note of Pauling's advocating possible bonding between large anions. Many examples of nonbonded or van der Waals interactions are considered from both points of view. The final section deals with the consequences of the realization that bonded quantum atoms that share an interatomic surface do not "overlap". The time has come for entering students of chemistry to be taught that the electron density can be seen, touched, and measured and that the chemical structures they learn are in fact the tracings of "bonds" onto lines of maximum density that link bonded nuclei. Matter, as we perceive it, is bound by the electrostatic force of attraction between the nuclei and the electron density.

Definition of molecular structure: by choice or by appeal to observation?

Determination of electronic wave functions from electronic densities is discussed. A systematic new method based on the constrained‐search formulation of density functional theory is developed, and the method is tested for Be, Ne, and Ar. The results show that the Kohn–Sham kinetic energy and the Kohn–Sham orbitals can be obtained readily and accurately by this method. Since Kohn–Sham orbitals are very close to Hartree–Fock orbitals, the problem of finding a physical meaningful wave function from an electron density is thereby solved.

Constrained‐search method to determine electronic wave functions from electronic densities

The method of the one-electron density matrix determination from X-ray diffraction data is developed. It takes into account the translational symmetry and is applicable to covalent crystals. The crystal energies and directed Compton profiles are calculated in impulse approximation for silicon and diamond. The agreement with the experiments is rather good.



Die Methode der Ein-Elektronen-Dichtematrixbestimmung aus Rontgenbeugungsdaten wird entwickelt. Sie berucksichtigt die Translationssymmetrie und ist auf kovalente Kristalle anwendbar. Die Kristallenergien und gerichteten Compton-Profile werden in Impulsnaherung fur Silizium und Diamant berechnet. Die Ubereinstimmung mit den Experimenten ist recht gut.

The Crystal Electron Energy and Compton Profile Calculations from X-Ray Diffraction Data

The electron density distribution and Bi atom displacements in Ba 0.87 K 0.13 BiO 3 perovskite single crystal are studied using both X-ray diffraction data and modified statistical method calculations. A noticeable covalent component in the Bi bond with neighboring oxygens is revealed. The nature of the Bi atom anharmonic displacements reflect either dynamic or static instability in its deal crystallographic position. The results obtained indirectly support the interpretation of the nature of incommensurate modulation in Ba 1− x K x BiO 3 crystals under electron irradiation.

Chemical bonding and Bi atom anharmonicity in Ba0.87K0.13BiO3 crystal

The role of the electron density distribution of the ground state in the theory, describing the structure and properties of the multielectron systems, is considered. Modern methods of deducing the electron density from X-ray coherent scattering data are presented. Examples of studying crystal properties from the experimental information obtained are given. It is emphasized, that the combination of theoretical methods with the possibilities of experiment promises the development of new methods for solid state studies.

I. M. Reznik

Papers

The Crystal Electron Energy and Compton Profile Calculations from X-Ray Diffraction Data

Chemical bonding and Bi atom anharmonicity in Ba0.87K0.13BiO3 crystal

Electron density distribution in crystals : theory problems and experiment