Brillouin-Wigner Perturbation Theory Research Articles

Theory of a mixed-valent impurity

We present a perturbative theory for the thermodynamic properties of a mixed-valent impurity in a metal. The impurity has two ionic configurations ${f}^{n\ensuremath{-}1}$ and ${f}^{n}$ (nondegenerate and ${n}_{\ensuremath{\lambda}}$-fold degenerate, respectively) with energies ${\ensuremath{\epsilon}}_{0}$ and (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}+\ensuremath{\mu}$), the difference (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$) being small. They mix via hybridization with conduction electrons (matrix element ${V}_{\mathrm{kf}}$). We show that for $D>({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0})\ensuremath{\gtrsim}\ensuremath{-}{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}\mathrm{ln}(\frac{D}{{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}})$ a Brillouin-Wigner perturbation theory is convergent. Here $\ensuremath{\Delta}={|{V}_{\mathrm{kf}}|}^{2}\ensuremath{\rho}(\ensuremath{\mu})$ is the virtual level width and $2D$ is the conduction-electron bandwidth, $\ensuremath{\rho}(\ensuremath{\mu})$ being the density of states at the Fermi level. The expansion parameter is the inverse of the orbital degeneracy ${n}_{\ensuremath{\lambda}}$. Since this is large (6 to 8), the expansion is quite convergent, and the lowest-order theory is accurate. This is checked by calculation of higher-order terms for various values of (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$). In the above range of (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$) the $f$-electron number is seen to change from ($n\ensuremath{-}1$) to about $(n\ensuremath{-}1)+0.80$, so that there is a perturbative theory for a strongly-mixed-valent impurity. Hybridization stabilizes the singlet ${f}^{n\ensuremath{-}1}$ relative to ${f}^{n}$, the maximum stabilization energy (level shift) being approximately ${n}_{\ensuremath{\lambda}}\ensuremath{\Delta}\mathrm{ln}(\frac{D}{{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}})$ for ${\ensuremath{\epsilon}}_{0}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$. This singlet ground state has been obtained variationally by Varma and Yafet, and from renormalization-group arguments by Haldane, and by Krishnamurthy, Wilkins, and Wilson; the Brillouin-Wigner perturbation theory has been used earlier by Bringer and Lustfeld. However, the recognition of ($\frac{1}{{n}_{\ensuremath{\lambda}}}$) as an expansion parameter and the consequent simplification of the theory are new. Physical properties such as valence, susceptibility, and specific heat are calculated as a function of ($\frac{{k}_{B}T}{\ensuremath{\Delta}}$) for various values of (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$). A simple way of including the effect of alloying pressure is described. Many characteristic properties of metallic dilute and concentrated mixed-valent systems, such as the temperature dependence of valence, the positive ${T}^{2}$ slope of the low-temperature susceptibility $\ensuremath{\chi}(T)$, the broad maximum in it, the relation between $\ensuremath{\chi}(0)$ and the Curie-Weiss temperature of high-temperature susceptibility, are qualitatively explained and quantitatively characterized for the first time. The results are directly applicable to dilute and nondilute alloys. They can also be applied to concentrated perfect lattice systems except at the lowest temperatures where relatively small intersite coupling leads to a uniform Fermi-liquid ground state. The Kondo limit, i.e., the nearly-${f}^{n}$-valent singlet which occurs for $({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0})\ensuremath{\ll}\ensuremath{-}{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}$, is not described by the present theory.

Connections between perturbation theory and the unitary transformation methods of deriving effective Hamiltonians

AbstractThe connections between open shell Brillouin–Wigner perturbation theory and the Van Vleck unitary transformation formalisms for generating effective Hamiltonians are explored. An explicit expression is obtained relating the generator Ŝ of the unitary transformation eiŝ with the amplitudes to be found from perturbation theory. The “renormalization effects” needed to produce the explicit “orthogonal‐Hermitian” form of the effective Hamiltonian in perturbation theory are related directly to the generator of the unitary transformation. The conclusions reached previously by Jørgensen and Brandow regarding the identity of the effective Hamiltonians of the formalisms are explicitly verified for the case that the generator Ŝ satisfied the Kemble condition. The procedure suggests how the powerful techniques of perturbation theory can be used within the unitary transformation framework to guarantee properly renormalized wave functions.

Valence-core self-consistency in theA=17system

We develop a systematic approach to the calculation of self-consistency effects in core plus valence nucleon systems. A detailed calculation of the effective one-body Hamiltonian for A=17 in an 11h..cap omega.. model space is readily extrapolated to a 20h..cap omega.. model space which is sufficient to develop accurate tails for the single-particle wave functions. The effective one-body interaction is found using a Brillouin-Wigner type perturbation theory which eliminates folded diagrams and leads to a manifestly Hermitian interaction. A renormalized Brueckner calculation is performed for A=16 and the results are employed in a shell-model study of A=17. The self-consistent results still have a weak dependence on the initial unperturbed Hamiltonian. However, we find a single unperturbed Hamiltonian which yields a reasonable binding energy for A=16 and, except for a weak spin-orbit splitting, reasonable results for the lowest states in the A=17 system. This agreement is important for continued valence s- d shell studies.

Description of uniform many-particle systems in density operator function space

Assuming the ground state wavefunction, ψ 0, of a boson fluid is known, and writing the excited state wavefunctions in the form Fψ 0, a linear eigenvalue equation of the form HF = EF is obtained, where E 0 + E is the excited state energy, E 0 is the ground state energy, and H is a non-hermitian operator which depends in a simple way upon U ≡ ln ψ 0 2 instead of the potential energy function. An extremum principle is derived in terms of an auxiliary hermitian Hamiltonian operator, H′. The many-body boson plane-wave basis, ϱ n (k 1 … k n ) is used to express U in terms of its Fourier components (ordered conveniently in terms of the number of nonzero arguments), making it possible to calculate matrix elements of ovcirc| H and H′ in that basis. A perturbation theory similar to Brillouin-Wigner perturbation theory is developed for the non-hermitian eigenvalue problem. Nonorthogonal perturbation theory is developed for the correlated basis ϱ n ψ 0. The requirement that these two perturbation theories be consistent produces useful relationships between the components of U and the static structure functions of ψ 0. These relationships are shown to reduce to previous results in the extreme case of low density and weak interactions.

SCF MO CI perturbation calculations of inner shell and valence shell vertical ionization potentials

A CI method for calculating inner and valence shell vertical ionization potentials is presented. It is based on ab initio SCF MO calculations for the neutral closedshell ground state followed by CI perturbation calculations for the ground and ion states including all spin and symmetry adapted singly and doubly excited configurations with respect to the main configurations of the state of interest. The state energy is computed by performing a CI calculation for a set of selected configurations, and then adding the contributions of the remaining configurations as estimated by second order Brillouin-Wigner perturbation theory. The use of the same set of MO's for all states together with the CI perturbation method makes the method rather rapid. The numerical results are, in spite of the limited Gaussian basis sets used, in good agreement with experiment.

Equations-of-motion formulation of many-body perturbation theory

A formulation of many-body perturbation theory starting from the operator equation (H,Q+)= omega Q+ is presented. The method of solution is based on an operator scalar product, (X/Y)=Tr(X+Y), which allows the use of resolvent and partitioning techniques to establish Rayleigh-Schrodinger or Brillouin-Wigner perturbation theory for the excitation energy and excitation operator, Q+. The excitation operator contains all information about the two states involved in the transition. Specific results are given for removal or addition of an electron and for excitations of particle-hole type and comparison with the Green's function methods is made.

A diagrammatic construction of formal E-independent model hamiltonian

A diagrammatic construction of formal E-independent model interaction (i.e. without second-quantization formalism) is suggested. We start from the quasi-degenerate Brillouin-Wigner perturbation theory, in framework of which an E-dependent model hamiltonian is simply constructed. Applying the “E-removing” procedure to this E-dependent model hamiltonian, the E-independent formal model hamiltonian either hermitian or non-hermitian can be diagrammatically easily derived. For the formal E-independent model hamiltonian the separability theorem is proved, which can be profitable used for a rather “formalistic” construction of a many-body E-independent model hamiltonian containing only linked-connected diagrammatic terms.

Density dependence of the roton spectrum in liquidHe4

The density dependence of the roton spectrum in liquid $^{4}\mathrm{He}$ is calculated using Brillouin-Wigner perturbation theory. The trial ground-state wave function is an extended Jastrow function, including three-body factors. The three-body factors produce an improved density dependence over the pure Jastrow trial function. We note, however, that the amount of improvement depends upon the particular approximation chosen for the three-particle structure function, ${S}^{(3)}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3})$. Specifically, better agreement with experiment is obtained by using an approximation for ${S}^{(3)}$ which has been obtained recently in the density-phase variable theory of the weakly interacting Bose gas, when compared to the convolution approximation for ${S}^{(3)}$.

Resolvent technique and Padé approximants in configuration interaction calculations

AbstractA perturbation approach based on resolvent technique and Padé approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Padé approximants of type II. The method is tested on a CI calculation of the energy of the lowest 1Σ state of the B2 molecule.

Long-wavelength-phonon spectrum in liquidHe4

The energy spectrum of elementary excitations in liquid $^{4}\mathrm{He}$ is studied in the long-wavelength limit by means of the Brillouin-Wigner (BW) perturbation theory for two-phonon processes. Calculation of an additional second-order BW energy correction to the Bijl-Feynman excitation spectrum is carried out by considering contributions from the leading correction to the convolution approximation for the three-particle distribution function. The resulting correction is found to be positive and about 7.9% of the second-order perturbation energy obtained previously with the convolution approximation.

On the existence of Stark ladders in finite crystals

The electronic states of a finite crystal in a uniform electric field are studied for a linear, one-band, tight-binding chain with nearest neighbour overlaps, the most favourable case for the existence of a Stark ladder in the energy spectrum. The use of Brillouin-Wigner perturbation theory leads directly to an exact ladder for an infinite chain and strongly suggests that a fraction of the levels arising from atoms in the central part of a finite chain form a weakly perturbed 'quasi-ladder'. The properties of continued fractions enable analysis of finite size effects on the Stark levels of isolated atoms and yield accurate upper and lower bounds for the edges of the perturbed energy spectrum. Detailed numerical calculations substantiate these conclusions.

Comparison of Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Theories

Parallel derivations of Rayleigh-Schrödinger and Brillouin-Wigner perturbation theory are presented, using the Dirac formalism. The relative advantages and disadvantages of the two are compared with the goal of smoothing the transition between elementary quantum mechanics and advanced approximation theory.

Variational Methods in the Wave Operator Formalism: Applications in Variation-Perturbation Theory and the Theory of Energy Bounds

Variational principles of the Lippmann-Schwinger type are used to develop approximations to eigenenergies and eigenfunctions within the wave-operator formalism. The present approach starts with exactly soluble ``primary'' eigenvalue equations to give explicit results valid beyond the limits of conventional perturbation theory. The variational functionals are expressed in terms of resolvents of the primary Hamiltonian, and bounds to the functionals are constructed also for cases where the resolvents are only partly known. Approximations to eigenenergies and eigenfunctions are obtained in terms of quantities in the Brillouin-Wigner perturbation theory. Connections with methods for upper and lower energy bounds are discussed, and the convergence properties of the nonlinear Padé summation is recovered in this way. Closed formulas within the double perturbation theory framework are presented as a logical extension.

Polaron zeeman effect of shallow donors in CdTe

Far infrared Fourier transform spectroscopy has been employed to study the Zeeman splitting of the 1 s to 2 p shallow donor impurity transition in CdTe. Polaron effects are observed and are quite well accounted for by Wigner-Brillouin perturbation theory.

A note on model surface state calculations

This letter discusses the application of Brillouin-Wigner perturbation theory to two models of the crystal potential used in surface state calculations.

Determinantal Method in Perturbation Theory

The determinantal method of deriving fundamental relations of a new perturbation theory, which the author has presented recently, is demonstrated. This method is simpler than the Green's function method which has been adopted in the previous paper. The Brillouin-Wigner perturbation theory is discussed for comparison.

Many-body basis for the optical model

A formal expression is derived for an optical model potential based on an assumed two-body interaction between nucleons, which provides an exact description of the elastic scattering of a single nucleon by a closed shell (or closed shell ±1) nucleus. A second-quantized description is used for the many-fermion system, the true state vector being expanded in terms of a complete set of single-particle model wave functions. Elimination of the variables of all but the scattered nucleon yields a weighting function which satisfies a one-body Schrödinger equation, whose S-matrix elements are identical with those of the true S-matrix between states corresponding to elastic scattering. The effective optical model potential is identified from this Schrödinger equation, and is found, of course, to be complex and non-local. It contains all the effects of the exclusion principle, and is in the form of a linked-cluster perturbation expansion, so that the spurious divergence of Brillouin-Wigner perturbation theory for a large number of nucleons is absent.