Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2105.05009 · Annales Henri Poincaré

Explicit Diagrammatic Solution of Normalised, Nondegenerate Rayleigh–Schrödinger Perturbation Theory

Joel C. Pommerening, David Peter Divincenzo

Published Jan 29, 2022• 0 citations
arXivHTMLPDF

Abstract

Abstract We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh–Schrödinger perturbation theory under the constraint that the approximate eigenvector is normalised to 1 in every order. The series can be expressed in terms of diagrams that were first introduced by Bloch (Nucl Phys 6:329, 1958) for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams’ topology to the coefficients.

DOI
10.1007/s00023-021-01150-0

Related papers

Perturbation theory without unperturbed solutions

R. F. Bishop, M. F. Flynn, Miloslav Znojil

A modified version of Rayleigh-Schr\"odinger (RS) perturbation theory, which has been developed previously, can dispense with the usual RS restriction that the unperturbed Hamiltonian should be diagonal in the chosen basis. This modified RS (MRS) perturbation scheme is discussed in some detail and extended to the degenerate case. The new degenerate MRS perturbation theory remains based upon a rearrangement of the unperturbed and perturbed parts of the Hamiltonian. Basically, an unperturbed Schr\"odinger equation is thereby produced which may be satisfied trivially and identically for an arbitrary Hamiltonian. The zeroth-order approximation to the energy remains a free parameter of the degenerate version of the MRS formalism. The explicit MRS expressions for the higher-order correction terms also preserve a strict analogy with their counterparts in the nondegenerate formalism. What is new is an enormous increase in the applicability and universality of the whole approach. Thus, a priori, the unperturbed Hamiltonian may now be chosen so as to have all of its larger off-diagonal elements much closer to those of the full original Hamiltonian than is permitted in the standard RS approach. This is a particular advantage of the whole MRS approach. By testing the method on two nontrivial model Hamiltonians, the method is demonstrated to have considerably improved convergence properties over the standard RS scheme, and at relatively little extra cost.

Physical Review A•May 1, 1989•17 citations

A coupled-cluster approach to the many-body perturbation theory for open-shell systems

Ingvar Lindgren

A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.

International Journal of Quantum Chemistry•Jun 18, 2009•148 citations

Diagrammatic Perturbation Theory

Jørgen Rammer

At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. This consists in dividing the Hamiltonian into one part representing a simpler well-understood problem and a nontrivial part, the effect of which is studied order by order. The expressions resulting from perturbation theory quickly become unwieldy. This chapter deals with propagators. It introduces the retarded and advanced propagator and describes their perturbation theoretic structure in a potential in terms of diagrams. A convenient method of representing perturbative expressions by diagrams was invented by Feynman. Besides the appealing aspect of representing perturbative expressions by drawings, the diagrammatic method can also be used directly for reasoning and problem solving. The easily recognizable topology of diagrams makes the diagrammatic method a powerful tool for constructing approximation schemes as well as exact equations that may hold true beyond perturbation theory.

CRC Press eBooks•May 4, 2018•2 citations

Hypervirial-perturbation methods

John P. Killingbeck

This chapter gives a brief sketch of some parts of Rayleigh-Schrödinger perturbation theory, the author have written in detail about perturbation theory elsewhere and concentrate here on a few ideas which are directly useful in numerical work on a microcomputer. The intellectual puzzle of how to reduce everything to an energy calculation gets really interesting if we want to calculate a local quantity, e.g. the value of the (normalised) wavefunction at the origin. The renormalised series trick seems to give the results to greater accuracy and more easily. The program makes an essential use of arrays, but since the array subscripts on the Pet cannot be negative the subscripts have to be changed from their values in the theoretical equations. In many-body theory, then, the sum-overstates formalism can be useful for giving a semi-quantitative account of various important effects, and the so-called diagrammatic perturbation theory used in atomic theory leans heavily on that formalism.

CRC Press eBooks•Jan 18, 2018•0 citations

Circular Scale of Time as a Way of Calculating the Quantum-Mechanical Perturbation Energy Given by the Schrödinger Method

Stanisław Olszewski

The Schrodinger perturbation energy for an arbitrary order N of the perturbation has been presented with the aid of a circular scale of time. The method is of a recurrent character and developed for a non-degenerate quantum state. It allows one to reduce the inflation of terms necessary to calculate known from the Feynman’s diagrammatical approach to a number below that applied in the original Schrodinger perturbation theory.

Journal of Quantum Information Science•Jan 1, 2014•4 citations

The linked-diagram expansion of the ground state of a many-electron system: A time-independent derivation

Rolf Manne

A time-independent derivation of the linked-diagram expansion of the ground-state energy of a many-electron system is given for the case of a nondegenerate unperturbed system. The starting point is the Rayleigh-Schrödinger perturbation theory, and the derivation follows in many respects suggestions by Čížek and Paldus, including the use of the Frantz-Mills factorization theorem for the cancellation of unlinked and renormalization diagrams.

International Journal of Quantum Chemistry•Jun 18, 2009•6 citations
Abstract We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh–Schrödinger perturbation theory under the constraint that the approximate eigenvector is normalised to 1 in every order. The series can be expressed in terms of diagrams that were first introduced by Bloch (Nucl Phys 6:329, 1958) for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams’ topology to the coefficients.
A modified version of Rayleigh-Schr\"odinger (RS) perturbation theory, which has been developed previously, can dispense with the usual RS restriction that the unperturbed Hamiltonian should be diagonal in the chosen basis. This modified RS (MRS) perturbation scheme is discussed in some detail and extended to the degenerate case. The new degenerate MRS perturbation theory remains based upon a rearrangement of the unperturbed and perturbed parts of the Hamiltonian. Basically, an unperturbed Schr\"odinger equation is thereby produced which may be satisfied trivially and identically for an arbitrary Hamiltonian. The zeroth-order approximation to the energy remains a free parameter of the degenerate version of the MRS formalism. The explicit MRS expressions for the higher-order correction terms also preserve a strict analogy with their counterparts in the nondegenerate formalism. What is new is an enormous increase in the applicability and universality of the whole approach. Thus, a priori, the unperturbed Hamiltonian may now be chosen so as to have all of its larger off-diagonal elements much closer to those of the full original Hamiltonian than is permitted in the standard RS approach. This is a particular advantage of the whole MRS approach. By testing the method on two nontrivial model Hamiltonians, the method is demonstrated to have considerably improved convergence properties over the standard RS scheme, and at relatively little extra cost.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. This consists in dividing the Hamiltonian into one part representing a simpler well-understood problem and a nontrivial part, the effect of which is studied order by order. The expressions resulting from perturbation theory quickly become unwieldy. This chapter deals with propagators. It introduces the retarded and advanced propagator and describes their perturbation theoretic structure in a potential in terms of diagrams. A convenient method of representing perturbative expressions by diagrams was invented by Feynman. Besides the appealing aspect of representing perturbative expressions by drawings, the diagrammatic method can also be used directly for reasoning and problem solving. The easily recognizable topology of diagrams makes the diagrammatic method a powerful tool for constructing approximation schemes as well as exact equations that may hold true beyond perturbation theory.
This chapter gives a brief sketch of some parts of Rayleigh-Schrödinger perturbation theory, the author have written in detail about perturbation theory elsewhere and concentrate here on a few ideas which are directly useful in numerical work on a microcomputer. The intellectual puzzle of how to reduce everything to an energy calculation gets really interesting if we want to calculate a local quantity, e.g. the value of the (normalised) wavefunction at the origin. The renormalised series trick seems to give the results to greater accuracy and more easily. The program makes an essential use of arrays, but since the array subscripts on the Pet cannot be negative the subscripts have to be changed from their values in the theoretical equations. In many-body theory, then, the sum-overstates formalism can be useful for giving a semi-quantitative account of various important effects, and the so-called diagrammatic perturbation theory used in atomic theory leans heavily on that formalism.