Usage de l'algégbre de lie su(n) dans l'étude des systémes quantiques á n états. IV. Geometrie du domaine des vecteurs de cohérence

The density operator of an n level quantum system is known to be a positive semidefinite, hermitian operator of trace one. In a previous article we have established, through su(n) algebras, a formalism where density operators are built from coherence vectors in a n² ? 1 dimension, real, Euclidean space. The last two conditions are then automatically satisfied. Being positive semidefinite means a restriction to the domain of coherence vectors. In this article we clarify this domain and obtain several equivalent tests to know if a given vector is part of it.     

Classical and quantum dynamical regimes in the bound space projected dynamics of strongly driven H Rydberg atoms

The bound space projected dynamics of the one dimensional model of H Rydberg atoms subjected to strong microwave radiation exhibits three dynamical regimes:(i) perturbatively localized,(ii) chaotic,(iii) external field dominated. This classification holds classically as well as quantum mechanically. The spectral properties of the bound space projected dipole operator dominating regime(iii) are studied analytically. A semiclassical analysis shows that its eigenfunctions, projected on the unperturbed basis states |n〉 of the one dimensional model, decay liken ^?7/3.     

Electronic effect on the rotational energy of a diatomic molecule

The rotational hamiltonian for a diatomic molecule has been rederived from the total classical hamiltonian. This procedure directly introduces the effect of electronic motion which is ordinarily neglected in zero-order approximation. Kronig's rotational hamiltonian is discussed and shown to be an approximation of our findings. Our general result is then specialized to ¹Σ states, and the theory tested by calculating the observed fractional discrepancy between the experimentally determined H³⁵Cl energy level constant Y₀₂ and its predicted value from Dunham's theory. When all corrections are summed, the results are in good agreement with experiment.     

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.     

Derivation and pilot application of a scheme for intermediate Hamiltonians in Fock space

A Fock space formulation of the intermediate Hamiltonian approach is derived by introducing shift operators in the equations determining effective Hamiltonians in Fock space. A non-hermitian intermediate Hamiltonian is constructed from the Fock space Bloch equation. An alternative derivation, based on a similarity transformation expression, is presented providing access to hermitian intermediate Hamiltonians. In a pilot application, the potential curves of the two lowest¹ _g ⁺ states of the H₂ molecule are calculated demonstrating the applicability of the scheme.     

Time-dependent perturbation: a recursive generation of the transition amplitudes between bound states

We present a new expansion of the solution to the time-dependent Schrödinger equation it ∂U/∂t = {H₀ + V(t)}U. A complete set of eigenvectors of H₀ spanning the Hilbert space in which H₀ and V operate is partitioned in two subsets. Transition amplitudes from the first subspace to the second one are calculated by building an adequate series of intermediate representations with respect to the couplings which produce the transition from the model space into its orthogonal complement. These expansions yield a coupling matrix series V⁽ⁿ⁾ for which an iterative solution V⁽ⁿ⁾ → V⁽ⁿ⁺¹⁾ is derived. This solution leads to a recursive numerical treatment of the calculation of transition amplitudes. A simple example, concerning a harmonic oscillator under an intense laser field, is considered using a Fourier analysis of the perturbation.     

Electric dipole moment of BrCl

The electric dipole moment of the BrCl molecule has been determined by a study of the Stark effect on nine hyperfine components of the J = 1 ← 0 rotational transition lying at 9 GHz. A direct diagonalisation procedure of the energy matrix of the total hamiltonian has been used and the electric dipole moment derived for the ground vibrational state of BrCl is |μ_o| = 0.519 ± 0.004 D.     

The massey parameter expansion for the semiclassical scattering matrix

The Massey parameter expansion method for the semiclassical scattering matrix is proposed, using the Magnus formalism. The first Massey parameter, a unitary member of this expansion, is found for the interaction hamiltonian, depending on time, as exp(-τ) and τ^?n. Other possible applications of the proposed method are discussed.     

The Effective Hamiltonian and Electron Correlation Energy

An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective hamiltonian deviates form the actual one by the presence of acorrelation operator.The cprrelation operatop is given in an explicit form.     

Solution of large configuration mixing matrices arising in partitioning technique and applications to the generalized eigenvalue problem

A method is presented that can be used (a) to determine the several lowest eigenvalues and eigenvectors of large symmetric matrices, (b) to solve the generalized eigenvalue problem associated with energy-dependent operators, that arises in computations involving energy-dependent many-body Green's functions and in the evaluation of the true parameters of the effective valence shell hamiltonian, and (c) to directly evaluate the matrices associated with resolvent operators. The applicability to large configuration mixing calculations arises when the N-electron basis functions can be easily broken down to a few dominant configurations (the primary block) and their complement. Using the partitioning technique, the effective hamiltonian within the primary block is directly evaluated. The method is extended to evaluation of the dynamical polarizability tensor, which effectively contains the contributions from all of the eigenstates of a hamiltonian matrix, without the necessity of explicitly calculating its eigenvalues and eigenvectors.     

Complete TDA and RPA Calculations on the Electronic Transitions of Fullerene‐C60 in the CNDO/S and INDO/S Approximations*

Complete single‐excitation mixing calculations on the electronic transitions of the icosahedral C₆₀ molecule have been carried out with the Tamm–Dancoff approximation (TDA) and random‐phase approximation (RPA) schemes in the CNDO/S and INDO/S approximations. The complete space of 14,400 (1p–1h) pairs is partitioned into subspaces classified according to the irreducible representations of the Ih group. For this purpose, matrix representations of the group generators are obtained on a fixed set of basis functions and are used to construct the projection operators. Degenerate molecular orbitals in each energy level are symmetry‐adapted to these projection operators. Degenerate (1p–1h) pairs or singly excited configuration wave functions are similarly symmetrized. In addition, the Clebsch–Gordan coefficients are obtained and listed in an Appendix. The TDA and RPA equations are then solved for each irreducible representation separately. Both schemes with the projection operators and with the Clebsch–Gordan coefficients gave the same results as expected, indicating that the calculations were correctly done. The transition energies from the ground state 1¹A_g to low‐lying singlet and triplet excited states and the oscillator strengths for the allowed transitions (n¹T_1u–1¹A_g) are given in tables. A proper way to normalize is discussed for the eigenvectors of the RPA‐type matrix equation. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Fragmentation dynamics of ionized argon clusters: an effective potential model

A dynamical model based on an effective potential is employed to describe metastability and the related time scales for evaporation processes in ionized Argon clusters (Ar) _n ⁺ , withn from 6 up to 27. The effective interaction, and its dependence on the cluster size, is obtained by combining previous ab initio results and Monte Carlo simulations for the above systems with an assumed linear variation of the cluster volume with then number of monomers. The overall metastability of the excited clusters is linked to the possible local rotational ‘temperature’ of such species and the distributions of the ensuing lifetimes are analysed as function of cluster size and of different nucleation mechanisms. It is found that an unusually large range of lifetime values is obtained from the present modelling, in general accord with earlier experiments, and that the existence of large rotational barriers can markedly delay the dissociation of metastable species.     

Selection rules and line-strength factors for multiphoton transitions in gas phase molecular spectroscopy

Multiphoton transitions are discussed in terms applicable to experimental spectroscopy. A simple tensor operator for the equations governing multiphoton transitions in molecules is obtained. Selection rules for these processes are derived by symmetry considerations and procedures for calculation of rotational line-strength factors are given. For the general transition, the rovibronic n-photon selection rules can be determined by the symmetry product of a n-photon operator with the rovibronic wavefunctions. The line-strength factors are found by considering the integral over rotations of the nth rank tensor operators. The line-strength evaluations do not depend on exact knowledge of the vibronic overlap, to within a constant factor.     

Nuclear hyperfine splitting in the BX electronic band system of 127I2

The spacings between seven hyperfine components in the R(127) line of the 11-6 band of the B-X electronic transition of ¹²⁷I₂ are fitted, with a standard deviation of 17.3 kHz, by varying the nuclear quadrupole coupling constants eqQ′ and egQ′', the magnetic spin—rotation constant C_I and the tensor and scalar nuclear spin—spin coupling constants d′ and a′ in the hyperfine hamiltonian. The P(13) line of the 430 band is also analysed using an identical hamiltonian and a standard deviation of 6.25 kHz is obtained. No evidence for a magnetic octupole coupling is found to the precision of the data although this effect was invoked by Hackel et al. for the P(13) line.     

Representation of one-dimensional motion in a morse potential by a quadratic hamiltonian

A representation of the algebraic hamiltonian for the anharmonic Morse oscillator as a quadratic form, H = T1ω($\text{1}\text{2}$P² + $\text{1}\text{2}$Q²), where P and Q are operators is derived. The commutator of P and Q is an operator that tends to i (times the identity operator) in the harmonic limit. Coherent states and anharmonic normal modes for a linear triatomic molecule are discussed as potential applications.     

The Fourier transform and diode laser spectrum of the ν2 band of diazomethane

The infrared spectrum of the ν₂ band (NN stretching) of gaseous diazomethane at 2100 cm^?1 has been measured by means of an interferometer and a tunable diode laser spectrometer. For the first time the rotational J and K_a structure of this A-type parallel band has been resolved. Since the spectrum was found to be perturbed it was not possible to fit the upper state levels to an overall hamiltonian. Nine subbands have been analysed with the support of millimeter wave data for the ground vibrational state. Term values for the ν₂ = 1 vibrational state with K_a up to 5 have been obtained and subband origins, effective rotational constants B and centrifugal distortion constants D and H were determined for each K_a substate.     

Ab initio calculation of the effective valence shell hamiltonian of carbon: Simultaneous treatment of neutral and ion states

The n = 2 effective valence shell hamiltonian, $\text{H}$^v, of carbon is evaluated through second order using ³P Hartree—Fock orbitals (5s4p) with added d functions to provide results within a few percent of the spd convergence limits. The calculated $\text{H}$^v is employed to evaluate the n = 2 valence states of C, C^?, C⁺, C²⁺ and C³⁺ with an average deviation of the 21 excitation energies, ionization potentials and electron affinity from experimental values of 0.32 eV. Three-electron parts of $\text{H}$^v contribute substantially to a number of these excitation energies.     

Electric dipole moment of the diatomic if molecule

The electric dipole moment of the IF molecule has been determined by a study of the Stark effect on the hyperfine components of the J = 1 ← 0 rotational transition lying in the 17 GHz region. The production of molecules was by microwave discharge of I₂ and C₆F₁₀(CF₃)₂. A direct diagonalisation of the energy matrix of the total hamiltonian has been used and the electric dipole moment derived for the ground vibrational state of IF is |μ₀ |= 1.948(20)D.     

On the evaluation of the geometrical factors utilized in ligand polarization calculations

The utility of the Ligand polarization model in solving many physical problems in quantum mechanics has been appreciated among scientists during the last years. Problems such as electric dipole strength, vibronic electric dipole strength, optical activity calculations have been carried out within the framework of a dynamic coupling mechanism. Taking advantage of the irreducible tensor method put forward by Griffith in the case of molecular symmetry groups, both the molecular states and relevant operators can be classified in terms of irreducible representations of the molecular group in question, and therefore it is most convenient to express the relevant operators involved in any specific calculation in a symmetry adapted form. As a starting point, we may classify our molecular states and operators in the 0-rotation group and lower symmetry groups may also be studied by using simple correlation properties. Here we aim to deal with d-d and f-f type of transitions, and hence the 2² (electric quadrupole), 2⁴ (electric hexadecapole) and the 2⁶-multipoles are considered in some detail. We have adopted, the octahedral set of functions as given by Griffith to define the 2^itl (l = 2, 4, 6) multipoles and obtain the corresponding geometrical factors for the various irreducible representations.     

Quasi-homopolar levels of hydrocarbons with conjugate bonds: Series for projected π-electron operators

These are states that go over to the 2p_z states of the neutral atoms as the latter recede to infinity; they include the ground state and most of the lower excited states. Then Schrödinger's equation and the operators for the physical quantities may be projected on the space of spin functions. A method is given for calculating the projected hamiltonian and operators as a rapidly convergent series in the number of interacting centers. Pair interactions are shown to play the main part in the spin hamiltonian. The convergence is examined for the series for the momentum and spin-density operators. Schrödinger's equation with the spin hamiltonian then gives a complete solution of the problem; as in the valence-bond method, the task is facilitated by the fact that the subspaces of defined system spin may be distinguished in spin space. A method is given for selecting the states from the measured terms for the molecule. It is shown that all absorption lines corresponding to excitation of such states should be weak for alternant hydrocarbons.