Ground and Excited States from Ensemble Variational Principles
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910704918855680 |
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| author | Ding, Lexin Hong, Cheng-Lin Schilling, Christian |
| author_facet | Ding, Lexin Hong, Cheng-Lin Schilling, Christian |
| contents | The extension of the Rayleigh-Ritz variational principle to ensemble states $ρ_{\mathbf{w}}\equiv\sum_k w_k |Ψ_k\rangle \langleΨ_k|$ with fixed weights $w_k$ 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 $ρ_{\mathbf{w}}$ as well as the individual eigenstates $|Ψ_k\rangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-Δ{E}_{\mathbf{w}} \leq ΔQ \leq d_+ Δ{E}_{\mathbf{w}}$ 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 $d_{\pm}\equiv d_{\pm}^{(Q)}(\mathbf{w},\mathbf{E})$ provide valuable insights into the optimal choice of the auxiliary weights $w_k$ in practical applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12104 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ground and Excited States from Ensemble Variational Principles Ding, Lexin Hong, Cheng-Lin Schilling, Christian Quantum Physics Mathematical Physics Chemical Physics The extension of the Rayleigh-Ritz variational principle to ensemble states $ρ_{\mathbf{w}}\equiv\sum_k w_k |Ψ_k\rangle \langleΨ_k|$ with fixed weights $w_k$ 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 $ρ_{\mathbf{w}}$ as well as the individual eigenstates $|Ψ_k\rangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-Δ{E}_{\mathbf{w}} \leq ΔQ \leq d_+ Δ{E}_{\mathbf{w}}$ 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 $d_{\pm}\equiv d_{\pm}^{(Q)}(\mathbf{w},\mathbf{E})$ provide valuable insights into the optimal choice of the auxiliary weights $w_k$ in practical applications. |
| title | Ground and Excited States from Ensemble Variational Principles |
| topic | Quantum Physics Mathematical Physics Chemical Physics |
| url | https://arxiv.org/abs/2401.12104 |