On a variational method for determining excited state wave functions

Messmer, R. P.

doi:10.1007/bf00527113

Cited by 31 publications

(23 citation statements)

References 9 publications

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“…Various attempts exist in the literature to introduce minimization principles for excited states. The first idea was to minimize   2 H E    [9], whose implementation needed an approximate value of E [10] or other assumptions which did not always guarantee convergence to the correct wave function [11]. Some other early attempts turn the Schrödinger equation (T+V)Ψ = ΕΨ into an integral equation using the Green's function G=(Ε−Τ) -1 and adjust E so as to make the eigenvalue μ of / 1 VGV V        .…”

Section: Previous Attemptsmentioning

confidence: 99%

“…But DFT does not compute excited wave functions, and the standard Quantum-Chemistry methods, via truncated wave functions, can only approach the correct excited energy, but not the exact excited wave function: Infinitely many wave functions (orthogonal to given lower lying truncated approximants) have exactly the excited energy (because, for example, between the exact 1 st excited state ψ 1 and an arbitrary large normalized expansion Φ, supposed to approximate the 1 st excited state, the system of equations   implementation needed an approximate value of E [10] or other assumptions which did not always guarantee convergence to the correct wave function [11]. Some other early attempts turn the Schrödinger equation (T+V)Ψ = ΕΨ into an integral equation using the Green's function G=(Ε−Τ) -1 and adjust E so as to make the eigenvalue μ of / 1 VGV V        .…”

Section: Introductionmentioning

confidence: 99%

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Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

Bacalis¹

2016

JCM

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The computation of small concise and comprehensible excited state wave functions is needed because many electronic processes occur in excited states. But since the excited energies are saddle points in the Hilbert space of wave functions, the standard computational methods, based on orthogonality to lower lying approximants, resort to huge and incomprehensible wave functions, otherwise, the truncated wave function is veered away from the exact. The presented variational principle for excited states, F n , is demonstrated to lead to the correct excited eigenfunction in necessarily small truncated spaces. Using Hylleraas coordinates for He 1 S 1s2s, the standard method based on the theorem of Hylleraas -Unheim, and MacDonald, yields misleading main orbitals 1s1s' and needs a series expansion of 27 terms to be "corrected", whereas minimizing F n goes directly to the corect main orbitals, 1s2s, and can be adequately improved by 8 terms. F n uses crude, rather inaccurate, lower lying approximants and does not need orthogonality to them. This reduces significantly the computation cost. Thus, having a correct 1 st excited state ψ 1 , a ground state approximant can be immediately improved toward an orthogonal to ψ 1 function. Also higher lying functions can be found that have the energy of ψ 1 , but are orthogonal to ψ 1 . F n can also recognize a "flipped root" in avoided crossings: The excited state, either "flipped" or not, has the smallest F n . Thus, state average is unnecessary. The method is further applied via conventional configuration interaction up to three lowest singlet states of He. The problem and the purposeThe study of excited states is already imperative especially as it concerns reactions, after activation, of stable species, like CO 2 or alkanes. First principles studies can only be utilized in truncated Hilbert spaces. Unfortunately, the standard methods of computing excited states in truncated spaces, although perhaps adequate for the energy and for spectroscopy, may yield incorrect wave functions (perhaps with correct energy), misleading for desired proper excitations. Thus, a method is needed (such as the present demonstrated) to yield excited state truncated wave functions that are not veered away from the exact Hamiltonian eigenfunctions. The ability to extend the variational principle to any excited state (without knowledge of the lower-lying exact eigenfunctions) has long been proven to be an inherent property of the Hamiltonian. The excited state truncated wave function based on the standard method of the Hylleraas and Undheim / MacDonald (HUM) theorem, is in principle incorrect in a more fundamental manner than just being truncated: Its accuracy must be strictly less than the accuracy of the ground state truncated approximant. On the other hand, an energy minimization orthogonally to all lower approximants ["orthogonal optimization" (OO)] must lead to a wave function lying lower than, and veered away from, the exact. A minimization principle for excited electronic states of a nondegenerate ...

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Section: Previous Attemptsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

Bacalis¹

2016

JCM

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“…whose global minimum is the exact (excited) energy eigenstate with energy closest to ω. 1,20 Somewhat surprisingly, we found that even the aggressive approximation…”

Section: Lagrange Multiplier Formalismmentioning

confidence: 71%

A generalized variational principle with applications to excited state mean field theory

Shea¹,

Gwin²,

Neuscamman³

2019

Preprint

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We present a generalization of the variational principle that is compatible with any Hamiltonian eigenstate that can be specified uniquely by a list of properties. This variational principle appears to be compatible with a wide range of electronic structure methods, including mean-field theory, density functional theory, multi-reference theory, and quantum Monte Carlo. Like the standard variational principle, this generalized variational principle amounts to the optimization of a nonlinear function that, in the limit of an arbitrarily flexible wave function, has the desired Hamiltonian eigenstate as its global minimum. Unlike the standard variational principle, it can target excited states and select individual states in cases of degeneracy or near-degeneracy. As an initial demonstration of how this approach can be useful in practice, we employ it to improve the optimization efficiency of excited state mean field theory by an order of magnitude. With this improved optimization, we are able to demonstrate that the accuracy of the corresponding second-order perturbation theory rivals that of singles-and-doubles equation-of-motion coupled cluster in a substantially broader set of molecules than could be explored by our previous optimization methodology.

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“…which if evaluated exactly has its global minimum at the Hamiltonian eigenstate whose energy is closest to ω. 56,57 Of course, many other properties and functions of the wavefunction can also be useful in specifying the desired state through the vector d. For example, if we knew that it should ideally be orthogonal to another nearby state |Φ and should have a dipole moment µ (not to be confused with the weighted average parameter µ above) of about µ 0 , we might use…”

Section: A Casscf Ansatzmentioning

confidence: 99%

Applying generalized variational principles to excited-state-specific complete active space self-consistent field theory

Hanscam¹,

Neuscamman²

2021

Preprint

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We employ a generalized variational principle to improve the stability, reliability, and precision of fully excited-statespecific complete active space self-consistent field theory. Compared to previous approaches that similarly seek to tailor this ansatz's orbitals and configuration interaction expansion for an individual excited state, we find the present approach to be more resistant to root flipping and better at achieving tight convergence to an energy stationary point. Unlike stateaveraging, this approach allows orbital shapes to be optimal for individual excited states, which is especially important for charge transfer states and some doubly excited states. We demonstrate the convergence and state-targeting abilities of this method in LiH, ozone, and MgO, showing in the latter that it is capable of finding three excited state energy stationary points that no previous method has been able to locate.

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On a variational method for determining excited state wave functions

Cited by 31 publications

References 9 publications

Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

A generalized variational principle with applications to excited state mean field theory

Applying generalized variational principles to excited-state-specific complete active space self-consistent field theory

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