The application of analytical methods in degenerate perturbation theory

Makhanek, A. G.; Korol'kov, V. S.; Fedorov, Arkady

doi:10.1007/bf00625851

Cited by 4 publications

(9 citation statements)

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“…Therefore, we neglected the frequency ( I ) in the energy denominators of the matrix element sums and used the following replacement, : W where uiy) is the hydrogen-like radial wave function 4f of the shell, u $ ; ! ,~( [ -~,~, is a first-order approximate radial wave function determined from the solution of the perturbed Schrodinger equation [3]. Of course, exact summation may also be madc as in the case of [3,91 where the terms ho of (7) are taken into account.…”

Section: On the Choice Of Approximationsmentioning

confidence: 99%

“…,~( [ -~,~, is a first-order approximate radial wave function determined from the solution of the perturbed Schrodinger equation [3]. Of course, exact summation may also be madc as in the case of [3,91 where the terms ho of (7) are taken into account. I n this case, however, the calculation of the TPA cross-section is awfully complicated.…”

Section: On the Choice Of Approximationsmentioning

confidence: 99%

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Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂

Makhanek

Korol'kov

Yuguryan

1988

Physica Status Solidi (b)

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A calculation is made of the parity-forbidden two-photon absorption cross section of the transition between the lowest Stark sublevels for a tetragonal site Cc3+ (symmetry Ca,) in a cubic fluorite crystal as well a8 of polarizat.ion characteristics of a CaF, : Ce3+ monocrystal with different orientations ( k l ) LlOO], k 11 [llO], k 11 [ l l l ] , k is t.he wave vector of the incident laser beam). The calculated cross-sectional value (a, = 3.7 x m~-~ s) and the polarization dcpendences agree with experiment.

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Section: On the Choice Of Approximationsmentioning

confidence: 99%

Section: On the Choice Of Approximationsmentioning

confidence: 99%

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂

Makhanek

Korol'kov

Yuguryan

1988

Physica Status Solidi (b)

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“…[4] through [7]. These 0020-7608/77/00 12-097 1$01 .OO methods made it possible not only to simplify the calculations but also to expand considerably the number of problems, which may be solved now rather simply, and to obtain first-order correction wave functions in an explicit form in terms of spherical coordinates for the arbitrary nl-shell.…”

Section: Introductionmentioning

confidence: 99%

“…(2) An analytical solution and an explicit form of the first-order correction wave functions in terms of spherical coordinates for an arbitrary degenerated state nl can also be obtained in those cases where an actual perturbation operator ( Vl -rl, VZ -rk) might obscure the degenerated state wave functions [4] (calculation problems of the polarizabilities and removing the forbiddenness in electrodipolar transitions between the equally even states and probabilities of the two-photon transitions between the terms of 4f electron configuration).…”

Section: Introductionmentioning

confidence: 99%

Laplace transform method in two‐photon spectroscopy theory

Makhanek

Korol'kov

Fedorov

1977

Int J of Quantum Chemistry

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AbstractsThis paper formulates a number of problems of the two-photon spectral theory and characterizes the analytical methods of solving them as worked out by the authors. The efficiency and wide application of the Laplace transform method is demonstrated by the most typical examples of the cross section analytical calculations of two-photon processes, dynamical polarizability, and bremsstrahlung at the scattering.Dans cet article on formule un nombre de p r o b l h e s de la theorie spectrale deux photons et on caracterise les mithodes analytiques pour les resoudre, developpees par les auteurs. L'efficacitt de la mtthode de la transformee de Laplace est demontree par des exemples typiques de ca!culs analytiques de processus a deux photons, de polarisabilitts dynamiques et de bremsstrahlung.In diesem Artikel werden gewisse Probleme der Spektraltheorie fur zwei Photonen formuliert und von den Verfassern entwickelte analytische Methoden fur ihre Losung charakterisiert. Die Leistungsfahigkeit und Anwendung der Methode mit Laplacetransformen wird demonstriert durch typische Beispiele von analytischen Berechnungen fur Zweiphotonenprozesse, dynamische Polarisierbarkeiten und Bremsstrahlung.

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“…The problem of computing the sum over the whole spectrum of intermediate states was solved by APT for the one-electron problem [16] using a Coulombic Green function (CGF), which is a solution of the following equation:…”

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confidence: 99%

Calculation of physical properties of atoms and ions based on analytical perturbation theory

Tryhuk

Феранчук

2011

J Appl Spectrosc

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Analytical perturbation theory is adapted to calculate properties of multi-electron atoms and ions. Results from zero-and first-order approximations are presented analytically and have an accuracy comparable to that of the Hartree-Fock method. It is shown for the first time that the summation over two-electron intermediate states in calculations of correlation corrections can be done in closed form.Introduction. The history of methods for solving the Schroedinger equation (SE) for a system of N nonrelativistic electrons in the field of a nucleus with charge Z is almost as long as that of quantum mechanics itself. Nevertheless, calculation of the whole spectrum of states of such a system with N ≥ 2 to any required accuracy remains an open question despite the increase in computing power. Results of recent studies [1] attest to its urgency. They focused on development of methods for direct numerical solution of the SE for the ground state of the He atom. There currently exists a large number of approximation methods that satisfy mainly the demands of spectroscopy, condensedstate physics, or quantum chemistry. However, the problem of calculating accurately spectra of multi-electron systems is still important in principle. Approximation methods for solving the SE for such systems include the statistical Thomas-Fermi model (TFM) [2-5] and density-functional theory based on it, the Hartree-Fock method (HFM) [6], and other variational approaches. In particular, the energy of the ground and low-lying excited states can be calculated to very high accuracy for two-electron atoms and ions by selecting a trial wave function with a sufficient number of parameters [7,8]. However, the ability to generalize this approach to multi-electron systems seems problematical.The regular method for solving the SE for atoms and ions is perturbation theory, which is based on an expansion over powers of Z -1 [9]. However, its zero-order approximation is insufficiently accurate because it does not at all take into account electron coupling. A problem arises in calculating the subsequent corrections that is related to computation of sums of the two-electron coupling operator matrix elements over the spectrum of intermediate states.This can be done only approximately and for rather simple systems using the variational principle.Herein analytical perturbation theory (APT) is developed. It can be used to overcome largely these difficulties. The zero-order approximation of APT enables the properties of atoms and ions to be calculated with accuracy comparable to that of the HFM. Moreover, the expressions for the subsequent corrections can be written in a closed form that is suitable for calculations using existing numerical methods.Model Hamiltonian of Zero-Order APT. The initial Hamiltonian of the examined system has the form (Coulombic units are used [10]):

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The application of analytical methods in degenerate perturbation theory

Cited by 4 publications

References 6 publications

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂

Laplace transform method in two‐photon spectroscopy theory

Calculation of physical properties of atoms and ions based on analytical perturbation theory

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The application of analytical methods in degenerate perturbation theory

Cited by 4 publications

References 6 publications

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E3/2(4f)−E3/2(5d) for Ce3+ in CaF2

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E3/2(4f)−E3/2(5d) for Ce3+ in CaF2

Laplace transform method in two‐photon spectroscopy theory

Calculation of physical properties of atoms and ions based on analytical perturbation theory

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Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂

Analysis of the Intensity and Polarization Dependence of Two‐Photon Absorption in the Channel E_3/2(4f)−E_3/2(5d) for Ce³⁺ in CaF₂