If Truncated Wave Functions of Excited State Energy Saddle Points Are Computed as Energy Minima, Where Is the Saddle Point?

Bacalis, Naoum C.

doi:10.1007/978-981-15-0006-0_13

Cited by 3 publications

(8 citation statements)

References 79 publications

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“…The k th excited state forms a pair of index- k saddle points that are also related by a sign-change in the wave function. The saddle-point nature of exact excited states was previously derived in the context of MC-SCF theory − and has been described by Bacalis using local expansions around an excited state . In fact, the Hessian index has been suggested as a means of targeting and characterizing a particular MC-SCF excited state. , In contrast, here the stationary properties of exact excited states have been derived using only the differential geometry of functions under orthogonality constraints.…”

Section: Exact Electronic Energy Landscapementioning

confidence: 99%

“…These stationary properties have been identified using the exponential parametrization of MC-SCF calculations 53 − 56 , 67 and can also be derived using local expansions around an exact eigenstate. 68 , 69 However, questions remain about the global structure of the exact energy landscape and the connections between the exact excited states.…”

Section: Introductionmentioning

confidence: 99%

“…It has long been known that the exact k th excited state forms a saddle point of the energy with k negative Hessian eigenvalues (where k = 0 is the ground state). These stationary properties have been identified using the exponential parametrization of MC-SCF calculations − , and can also be derived using local expansions around an exact eigenstate. , However, questions remain about the global structure of the exact energy landscape and the connections between the exact excited states.…”

Section: Introductionmentioning

confidence: 99%

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Energy Landscape of State-Specific Electronic Structure Theory

Burton

2022

J. Chem. Theory Comput.

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State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree–Fock and excited-state mean-field representations of singlet H 2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.

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Section: Exact Electronic Energy Landscapementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Energy Landscape of State-Specific Electronic Structure Theory

Burton

2022

J. Chem. Theory Comput.

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show abstract

“…The saddle-point nature of exact excited states was previously derived in the context of MC-SCF theory, [53][54][55][56] and has been described by Bacalis using local expansions around an excited state. 69 In fact, the Hessian index has been suggested as a means of targeting and characterising a particular MC-SCF excited state. 54,55 In contrast, here the stationary properties of exact excited states have been derived using only the differential geometry of functions under orthogonality constraints.…”

Section: Properties Of Exact Stationary Pointsmentioning

confidence: 99%

“…These stationary properties have been identified using the exponential parametrisation of MC-SCF calculations [53][54][55][56]67 and can also be derived using local expansions around an exact eigenstate. 68,69 However, questions remain about the global structure of the exact energy landscape and the connections between exact excited states.…”

Section: Introductionmentioning

confidence: 99%

Supporting Notebook for "Energy Landscape of State-Specific Electronic Structure Theory"

Burton

2021

View full text Add to dashboard Cite

State-specific approximations can provide an accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and Excited-State Mean-Field representations of singlet H 2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimisation landscape and elucidate the challenges faced by variance optimisation algorithms, including the presence of unphysical saddle points or maxima of the variance.

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Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory

Marie

Burton

2023

J. Phys. Chem. A

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State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed-and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higherenergy excited states in H 2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet−triplet crossing in CH 2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

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If Truncated Wave Functions of Excited State Energy Saddle Points Are Computed as Energy Minima, Where Is the Saddle Point?

Cited by 3 publications

References 79 publications

Energy Landscape of State-Specific Electronic Structure Theory

Energy Landscape of State-Specific Electronic Structure Theory

Supporting Notebook for "Energy Landscape of State-Specific Electronic Structure Theory"

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory

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