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arXiv:2411.14490

On the application of the Rayleigh-Ritz method to a projected Hamiltonian

Francisco Marcelo Fernández

Published Nov 20, 2024• 0 citations
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Abstract

We apply the well known Rayleigh-Ritz method (RRM) to the projection of a Hamiltonian operator chosen recently for the extension of the Rayleigh-Ritz variational principle to ensemble states. By means of a toy model we show that the RRM eigenvalues approach to those of the projected Hamiltonian from below in most cases but a few ones. We also discuss the effect of an energy shift and the projection of the identity operator.

DOI
10.48550/arxiv.2411.14490

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We apply the well known Rayleigh-Ritz method (RRM) to the projection of a Hamiltonian operator chosen recently for the extension of the Rayleigh-Ritz variational principle to ensemble states. By means of a toy model we show that the RRM eigenvalues approach to those of the projected Hamiltonian from below in most cases but a few ones. We also discuss the effect of an energy shift and the projection of the identity operator.
The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues E1{E}_{1}E1​\ensuremath{\le}E2{E}_{2}E2​\ensuremath{\le}...\ensuremath{\le}EM{E}_{M}EM​ of a Hamiltonian H, and given M real numbers w1{w}_{1}w1​\ensuremath{\ge}w2{w}_{2}w2​\ensuremath{\ge}...\ensuremath{\ge}wM{w}_{M}wM​>0, an upper bound for the weighted sum w1{w}_{1}w1​E1{E}_{1}E1​ +w2{w}_{2}w2​E2{E}_{2}E2​+...+wM{w}_{M}wM​EM{E}_{M}EM​ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (w1{w}_{1}w1​=w2{w}_{2}w2​=...=wM{w}_{M}wM​). Applications of the generalized principle are discussed.
The extension of the Rayleigh-Ritz variational principle to ensemble states ρw≡∑kwk|Ψk⟩⟨Ψk| with fixed weights wk lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state ρw as well as the individual eigenstates |Ψk⟩ and eigenenergies Ek . To be more specific, we derive linear bounds d−ΔEw≤ΔQ≤d+ΔEw on the errors ΔQ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of d±≡d±(Q)(w,E) provide valuable insights into the optimal choice of the auxiliary weights wk in practical applications.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.
We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.