Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration

Yao, Chen; Bulik, Ireneusz W.; Jiménez-Hoyos, Carlos A.; Henderson, Thomas M.; Scuseria, Gustavo E.

doi:10.1063/1.4824905

Cited by 28 publications

(29 citation statements)

References 41 publications

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“…This is demonstrated by the coalescence of the low energy UHF states with the ground state RHF solution at the Coulson-Fischer point in H 2 , 34 although further examples are observed in a wide array of molecular systems. [35][36][37] To construct a continuous basis set of SCF solutions for NOCI, Thom and Head-Gordon proposed that Hartree-Fock states may need to be followed into the complex plane. 11 We have recently reported a holomorphic Hartree-Fock theory as a method for constructing a continuous basis of SCF determinants in this manner.…”

Section: Introductionmentioning

confidence: 99%

Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems

Burton

Gross

Thom

2018

J. Chem. Theory Comput.

107

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We explore the existence and behaviour of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with n basis functions is rigorously identified as 1 2 (3 n − 1), proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO-3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH 2+and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the π electrons as a twoelectron problem, and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.

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Section: Introductionmentioning

confidence: 99%

Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems

Burton

Gross

Thom

2018

J. Chem. Theory Comput.

107

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“…This is a point that has been very clearly brought to the quantum chemical community in recent years by the Scuseria group (see Refs. [ ] and citations therein). Because symmetry breaking is the by‐product of a mean‐field description, it can be considered artifactual.…”

Section: Introductionmentioning

confidence: 99%

Current development of noncollinear electronic structure theory

Goings

Egidi

2017

Int J of Quantum Chemistry

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In this Perspective, we review recent developments in noncollinear electronic structure theory.After a brief historical overview of studies into broken symmetry wave functions, we show that noncollinear wave functions are necessary for studying spin and magnetic phenomena on account of spin-symmetry breaking terms in the Hamiltonian. Recent developments applying noncollinear electronic structure theory to magnetization dynamics, spin dynamics, and spin-orbit coupling in excited state properties are showcased. We also discuss some recent developments in noncollinear density functional theory. Finally, we comment on the future of noncollinear electronic structure theory.noncollinear spin, relativistic electronic structure theory, symmetry breaking, time-dependent electronic structure theory, two-component method 1 | I N TR ODU C TI ON Mean-field, and self-consistent field (SCF), methods are the backbone of modern electronic structure theory. Since the introduction of the singledeterminant ansatz by Hartree and Fock, there has been a great deal of work on understanding the nature of the Hartree-Fock (HF) solutions as well as how to improve their utility in the description of atoms and molecules. It was recognized early on that the solution of the HF equations required a balance between two (sometimes competing) goals. [1][2][3][4][5] The first was that the HF equations would be at a variational minimum: the best SCF solution would be the lowest energy single-determinant approximation to the full configuration interaction (FCI) equations (for a given basis).Solutions of the HF equations would always seek to find the lowest possible energy solution, so as to satisfy the variational principle. Conversely, it was known that the underlying electronic Hamiltonian is invariant to several unitary transformations, most notably spin rotations. This invariance expresses itself as a conservation law, and the conserved quantity is total spin angular momentum S 2 as well as the spin projection along an arbitrary axis, S z . Surely a physically meaningful solution of the HF approximation would also have good quantum numbers, so invariance to rotations in the spin manifold was built-in to the HF equations as a constraint in the single determinantal expression. For example, the restricted Hartree-Fock (RHF) equations contain only paired electrons in orbitals, yielding solutions with good spin quantum numbers.Unfortunately, it was quickly found that there are many cases where insisting that the HF solution be a spin eigenstate yields energies that are higher than they would otherwise be if spin symmetry constraints were relaxed. For example, in the classic example of the dissociation of H 2 one finds that when the bond is stretched to a certain distance (the Coulson-Fischer point) an RHF to UHF instability forms and qualitatively correct dissociation curve are obtained only in the spin symmetry broken state. The spin symmetry breaking arises when the solutions approach a (near) degeneracy. Hartree-Fock solutions are unstable...

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“…Generalized Hartree Fock (GHF) should be used. GHF has been researched for a long time, [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and while it has only rarely been used, it is gaining momentum. For example, GHF has recently been used to obtain interesting new perspectives on a variety of fullerenes.…”

Section: Introductionmentioning

confidence: 99%

A simple way to test for collinearity in spin symmetry broken wave functions: General theory and application to generalized Hartree Fock

Small

Sundstrom

Head‐Gordon

2015

The Journal of Chemical Physics

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We introduce a necessary and sufficient condition for an arbitrary wavefunction to be collinear, i.e., its spin is quantized along some axis. It may be used to obtain a cheap and simple computational procedure to test for collinearity in electronic structure theory calculations. We adapt the procedure for Generalized Hartree Fock (GHF), and use it to study two dissociation pathways in CO2. For these dissociation processes, the GHF wave functions transform from low-spin Unrestricted Hartree Fock (UHF) type states to noncollinear GHF states and on to high-spin UHF type states, phenomena that are succinctly illustrated by the constituents of the collinearity test. This complements earlier GHF work on this molecule.

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Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration

Cited by 28 publications

References 41 publications

Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems

Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems

Current development of noncollinear electronic structure theory

A simple way to test for collinearity in spin symmetry broken wave functions: General theory and application to generalized Hartree Fock

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