Ground and Excited States from Ensemble Variational Principles
Lexin Ding, Cheng-Lin Hong, Christian Schilling
Abstract
The extension of the Rayleigh-Ritz variational principle to ensemble states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x03C1;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">w</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2261;</mml:mo><mml:munder><mml:mo>&#x2211;</mml:mo><mml:mi>k</mml:mi></mml:munder><mml:msub><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">&#x03A8;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo><mml:mo fence="false" stretchy="false">&#x27E8;</mml:mo><mml:msub><mml:mi mathvariant="normal">&#x03A8;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:math> with fixed weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x03C1;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">w</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> as well as the individual eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">&#x03A8;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo></mml:math> and eigenenergies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>. To be more specific, we derive linear bounds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>d</mml:mi><mml:mo>&#x2212;</mml:mo></mml:msub><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>E</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">w</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>Q</mml:mi><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>E</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">w</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> on the errors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>Q</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x00B1;</mml:mo></mml:mrow></mml:msub><mml:mo>&#x2261;</mml:mo><mml:msubsup><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x00B1;</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">w</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">E</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math> provide valuable insights into the optimal choice of the auxiliary weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> in practical applications.