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Physical Review A

Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

Eberhard K. U. Gross, L. Nunes de Oliveira, Walter Kohn

Published Apr 1, 1988• 352 citations
Publisher

Abstract

The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues ${E}_{1}$\ensuremath{\le}${E}_{2}$\ensuremath{\le}...\ensuremath{\le}${E}_{M}$ of a Hamiltonian H, and given M real numbers ${w}_{1}$\ensuremath{\ge}${w}_{2}$\ensuremath{\ge}...\ensuremath{\ge}${w}_{M}$>0, an upper bound for the weighted sum ${w}_{1}$${E}_{1}$ +${w}_{2}$${E}_{2}$+...+${w}_{M}$${E}_{M}$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (${w}_{1}$=${w}_{2}$=...=${w}_{M}$). Applications of the generalized principle are discussed.

DOI
10.1103/physreva.37.2805

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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 {w}_{1}$${E}_{1} +{w}_{2}$${E}_{2}+...+{w}_{M}$${E}_{M} 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 maximum-entropy principle is used to generalize the variational principle for the ground state (Rayleigh-Ritz minimization principle) to ensembles of unequally weighted states. A hierarchy of nested inequalities for the associated ``free energies'' is deduced. Applications of these inequalities to the estimation of the highest multiplet energy and a generalization of density-functional theory to ensembles of quantum states are discussed. Previous results given in the literature are obtained as special cases of our results under isothermal and adiabatic optimizations.
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.
We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<12|α| < \tfrac{1}{2}∣α∣<21​ for generalized Gaussian trial functions ψ(z)=eαz2+βzψ(z) = e^{αz^2 + βz}ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−12∂x2+12x2+λx4\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4H^=−21​∂x2​+21​x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=znψ_n(z) = z^nψn​(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+12+3λ4(2n2+2n+1)E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)En​=n+21​+43λ​(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4x^3 + x^4x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈12+3μ4−9λ24+⋯E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdotsE0​≈21​+43μ​−49λ2​+⋯).