Renormalized Rayleigh-Schrödinger perturbation theory

One method which has been used in the literature to determine the eigenvalues F _k ⁽⁰⁾ of a hamiltonian operator = + W ( = kinetic energy operator) is to apply an approximation scheme (e.g. variational method) to the operator _R () = + V + (W – V) with eigenvalues G _k (), where the eigenvalue problem associated with ^(o) = + V is solvable. Specifically, F _k ⁽⁰⁾ = G _k (1). We investigate the method from a perturbation theoretic viewpoint. There is a renormalization map R:, [0, 1], [0, ), which relates the G() to the eigenvalues E() of () = + V + W. This, in turn, implies a linear relationship between the Rayleigh-Schrödinger -series coefficients G ⁽ⁿ⁾ and the -series coefficients E ⁽ⁿ⁾ of the form G=CE, where C is an infinite lower-triangular matrix. The renormalized -series, 01, is useful in the accurate computation of F ⁽⁰⁾ as well as the eigenvalues E(), 0< . In standard cases, the -series is Borel summable to G(). Applications are made to anharmonic oscillators and hydrogen atoms in radial fields.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Iterative degenerate perturbation theory

A fairly general degenerate perturbation theory has been considered which involves two versions, i.e., the Rayleigh-Schrodinger and Brillouin-Wigner treatments. The simple recursive formulas have been found for effective Hamiltonians which allow one to split the degenerate energy levels in any orders of this theory.     

Adiabatic perturbation theory for the degenerate case. II. Perturbation of degenerate bound states embedded in the continuum

An adiabatic formula for the contracted Hamiltonian in a reference space containing bound-state eigenfunctions of degenerate energy levels embedded in the continuum is derived. A general factorization theorem for the dynamic operatorS_α(0, – ∞/λ) is proved, and the cancellation of the pole singularities in the perturbation series of the contracted Hamiltonian in adiabatic form is discussed.     

Generalized density functional theory for degenerate states

An extension of density functional theory is proposed for degenerate states. There are suitably selected basic variables beyond the subspace density. Generalized Kohn-Sham equations are derived. A direct method is proposed to ensure the fixed value of ensemble quantities. Then the Kohn-Sham equations are similar to the conventional Kohn-Sham equations. But the Kohn-Sham potential is different for different ensembles. A simple local expression is proposed for the correlation energy.     

The perturbation theory for non-degenerate states and the extended Hückel method

A formal study of the perturbation theory of the pseudosecular equations and particularly of those within the framework of the EHT method, proposed by Hoffmann, is given. In the case of the first-order perturbation in the hamiltonian matrix, it is shown that the classical and the present formalisms are equivalent. In the other more general cases, the recurrence formulae in the n-th order for the eigenvalues and eigenvectors are also given.

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Zusammenfassung Die Störungsrechnung für Pseudo-Säkulärgleichungen, insbesondere für diejenigen, die in der EHT Methode von Hoffmann auftreten, wird untersucht. Im Falle einer Störung erster Ordnung des Hamiltonoperators sind der klassische und der vorliegende Formalismus gleich. In den allgemeineren Fällen werden die Rekursionsformeln n-ter Ordnung für die Eigenwerte und Eigenvektoren gegeben.

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Résumé Etüde formelle de la théorie perturbative des équations pseudo séculaires, en particulier de celles de la méthode EHT de Hoffmann. Au premier ordre de perturbation de la matrice hamiltonienne, le formalisme classique équivaut à celui proposé ici. Dans les autres cas on donne les formules de récurrence à l'ordre n pour les valeurs et les vecteurs propres.

     

On the relation between Peierls' perturbation theory for projected states and exchange perturbation theory

The derivations of a rather general class of exchange perturbation theories (or symmetry-adapted perturbation theories) and Peierls' perturbation theory for projected states are discussed. While the approaches are different, the equivalence of the resulting formalisms is established.     

Adiabatic perturbation theory for the degenerate case. I. Perturbation of isolated degenerate or quasidegenerate levels

Adiabatic formulae for secular operators and contracted Hamiltonians in an arbirary combination of degenerate or quasidegenerate subspaces are derived. A detailed consideration of the adiabatic limit in the power series is given, and “stability” of proper linear combinations with respect to a transformation produced by the S_α-matrix is proved.     

Molecular-orbital-free algorithm for excited states in time-dependent perturbation theory

A nonlinear conjugate gradient optimization scheme is used to obtain excitation energies within the random phase approximation (RPA). The solutions to the RPA eigenvalue equation are located through a variational characterization using a modified Thouless functional, which is based upon an asymmetric Rayleigh quotient, in an orthogonalized atomic orbital representation. In this way, the computational bottleneck of calculating molecular orbitals is avoided. The variational space is reduced to the physically-relevant transitions by projections. The feasibility of an RPA implementation scaling linearly with system size N is investigated by monitoring convergence behavior with respect to the quality of initial guess and sensitivity to noise under thresholding, both for well- and ill-conditioned problems. The molecular-orbital-free algorithm is found to be robust and computationally efficient, providing a first step toward large-scale, reduced complexity calculations of time-dependent optical properties and linear response. The algorithm is extensible to other forms of time-dependent perturbation theory including, but not limited to, time-dependent density functional theory.     

Diagrammatic analysis of adiabatic and time-independent perturbation theory for degenerate energy levels

Time-ordered folded diagrams are used to represent the effective hamiltonian in the adiabatic formalism. Resummation of the diagrams is shown to give a term-by-term correspondence with time-independent perturbation theory.     

Perturbation theory for degenerate states of atomic and molecular systems

The first few terms of the perturbation expansions for the function and the energy shift of a degenerate state of an arbitrary quantum mechanical system are obtained using the adiabatic formula. It is shown how the expansion for the secular operator may be obtained from the expansion for the function. The results are used to calculate energies of the ground and some excited states and multiplet splittings of some beryllium-like ions.     

Degenerate states perturbation theory and continued fractions

An extension is made for degenerate levels of a recently proposed method for calculating higher-order terms of eigenvalues in a perturbation system with discrete energy levels. The method is used for calculating ground-state energy eigenvalue corrections for a rigid rotator which has a dipole ajid interacts with an applied electric field.     

Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory

Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used "average of degenerate states approximation" for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases.     

Möller-plesset perturbation theory for excited states of molecules

We consider the criteria that should be met by the perturbation theory for excited states to preserve the advantages of the Möller-Plesset version for the ground state. A few zero-approximation Hamiltonians are proposed and discussed. Expressions for the first-order corrections to the wave function and for the second-order corrections to the energy of excited states are obtained. The expressions are consistent with the conditions of orthogonality of states with the same symmetry. It is shown that the perturbation theory for excited states proposed in this work is identical to the Möller-Plesset perturbation theory for the ground state.     

Intruder states in multireference perturbation theory: the ground state of manganese dimer

A detailed analysis of a severe intruder state problem in the multistate multireference perturbation theory (MS-MRPT) calculations on the ground state of manganese dimer is presented. An enormous number of detected intruder states (> 5000) do not permit finding even an approximate shape of the X(1)Sigma(g) (+) potential energy curve. The intruder states are explicitly demonstrated to originate from quasidegeneracies in the zeroth-order Hamiltonian spectrum. The electronic configurations responsible for appearance of the quasidegeneracies are identified as single and double excitations from the active orbitals to the external orbitals. It is shown that the quasidegeneracy problem can be completely eliminated using shift techniques despite of its severity. The resultant curves are smooth and continuous. Unfortunately, strong dependence of the spectroscopic parameters of the X(1)Sigma(g) (+) state on the shift parameter is observed. This finding rises serious controversies regarding validity of employing shift techniques for solving the intruder state problem in MS-MRPT. Various alternative approaches of removing intruder states (e.g., modification of the basis set or changing the active space) are tested. None of these conventional techniques is able to fully avoid the quasidegeneracies. We believe that the MS-MRPT calculations on the three lowest A(g) states of manganese dimer constitute a perfect benchmark case for studying the behavior of MRPT in extreme situations.