The practical implementation of Löwdin's method for spin projection

Pons, Margalida

doi:10.1002/qua.25770

Cited by 7 publications

(25 citation statements)

References 34 publications

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“…In fact, Pons Viver demonstrated that the CO model greatly simplifies the expression of the EHF energy . In the present study, we do not refer to the configuration interaction part of Pons Viver's paper …”

Section: Introductionmentioning

confidence: 88%

“…In contrast, very recently, Pons Viver proposed the practical and exact implementation of Löwdin's spin projection operator (ie, the EHF method). The characteristic of Pons Viver's implementation is the practical use of the expectation value of Löwdin's spin projection operator.…”

Section: Introductionmentioning

confidence: 99%

“…The characteristic of Pons Viver's implementation is the practical use of the expectation value of Löwdin's spin projection operator. He denoted the expectation value as

scriptG ((), z)

, provided the practical and compact form of

scriptG ((), z)

, and showed that the total electronic energy of the EHF method is expressed in terms of

scriptG ((), z)

. Incidentally, Mayer et al also introduced the quantities

A_{b}^{a}

corresponding to

scriptG ((), z)

in order to express the EHF energy.…”

Section: Introductionmentioning

confidence: 99%

“…Incidentally, Mayer et al also introduced the quantities

A_{b}^{a}

corresponding to

scriptG ((), z)

in order to express the EHF energy. Another substantial idea of Pons Viver may be the application of the model of common orbitals (CO) (or closed‐shell orbitals) to the EHF method, although usually the CO model is adopted in the restricted open‐shell Hartree‐Fock (ROHF) method . In fact, Pons Viver demonstrated that the CO model greatly simplifies the expression of the EHF energy .…”

Section: Introductionmentioning

confidence: 99%

“…Another substantial idea of Pons Viver may be the application of the model of common orbitals (CO) (or closed‐shell orbitals) to the EHF method, although usually the CO model is adopted in the restricted open‐shell Hartree‐Fock (ROHF) method . In fact, Pons Viver demonstrated that the CO model greatly simplifies the expression of the EHF energy . In the present study, we do not refer to the configuration interaction part of Pons Viver's paper …”

Section: Introductionmentioning

confidence: 99%

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A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

Yoshizawa

2020

Int J of Quantum Chemistry

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Recently, the molecular electronic structure theories for efficiently treating static (or strong) correlation in a black-box manner have attracted much attention. In these theories, a spin projection operator is used to recover the spin symmetry of a broken-symmetry Slater determinant. Very recently, Pons Viver proposed the practical and exact implementation of Löwdin's spin projection operator (Int. J. Quantum Chem. 2019, 119, e25770). In the present study, we attempt to supply mathematical proofs to Pons Viver's proposals and show a condition for establishing Pons Viver's implementation. Moreover, we explicitly derive the (spin projected) extended Hartree-Fock (EHF) equations on the basis of the model of common orbitals (ie, closed-shell orbitals used in the restricted open-shell Hartree-Fock (ROHF) method), which was combined by Pons Viver with the EHF method. K E Y W O R D S extended Hartree-Fock method, extended pairing theorem, spin projection operator, static correlation 1 | INTRODUCTIONRecently, in the field of molecular electronic structure theory, Scuseria and co-workers have developed the projected Hartree-Fock (PHF) method in order to efficiently treat static (or strong) correlation in a black-box manner. [1][2][3][4] In the PHF method, to deliberatively break spin symmetry of a single Slater determinant and then variationally restore it, a spin projection operator such as Löwdin's spin projection operator [5] is employed. [1] Concretely, the projection operator is expressed as an integration of the spin-rotation operator (ie, Percus-Rotenberg integration [6] ) and it can be numerically evaluated with grid points. [3,7] As a result, one must determine the appropriate number of grid points, and then one must calculate the rotated Fock matrix at each grid point. [7] These results may give a negative image to the PHF method, but it has been reported that the PHF method has modest mean-field computational cost. [1][2][3] Moreover, according to Lestrange et al, [7] these inconveniences originating from the use of the grid points can be mitigated by using an efficient algorithm and Lebedev integration grids. On the other hand, unfortunately, the PHF method is not size consistent, [1] but Scuseria and co-workers have attempted to reduce the size consistency errors while exploring the origin of the errors. [8,9] On the other hand, to obtain a wave function of the desired spin multiplicity from an unrestricted Hartree-Fock (UHF) wave function, Mayer et al [10][11][12] already derived the (spin projected) extended Hartree-Fock (EHF) equations by using both Löwdin's spin projection operator and generalized Brillouin theorem [13][14][15] in the 1970s. It has been known that for small molecules, the EHF method yields good results, [12] but its computational cost is high due to its quite complicated equations. [16] Moreover, the EHF method is also not size consistent [16][17][18] and another serious shortcoming of the EHF method is that for large systems the EHF energy per particle reduces to

show abstract

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

“…The characteristic of Pons Viver's implementation is the practical use of the expectation value of Löwdin's spin projection operator. He denoted the expectation value as

scriptG ((), z)

, provided the practical and compact form of

scriptG ((), z)

, and showed that the total electronic energy of the EHF method is expressed in terms of

scriptG ((), z)

. Incidentally, Mayer et al also introduced the quantities

A_{b}^{a}

corresponding to

scriptG ((), z)

in order to express the EHF energy.…”

Section: Introductionmentioning

confidence: 99%

“…Incidentally, Mayer et al also introduced the quantities

A_{b}^{a}

corresponding to

scriptG ((), z)

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

Yoshizawa

2020

Int J of Quantum Chemistry

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Half‐Projected Hartree–Fock method: History and application to excited states of the same symmetry as the ground state

Ruiz

2022

Int J of Quantum Chemistry

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Spin projected wave functions are generalizations of the Hartree-Fock wave function. Among them, the Half-Projected Hartree-Fock (HPHF) wave function is a nearly pure wave function of spin and recovers a small part of the spin correlation energy. This paper reviews the history of the HPHF theory, not only from the conceptual point of view but also providing a compilation of the publications of this method over the years until now. In addition, the extension of the HPHF method to the calculation of excited states with the same symmetry as the ground state will be discussed. The possible variational collapse during the calculation of a singlet excited state of the same symmetry as the ground state is avoided by orthogonalizing a pair of corresponding orbitals, one occupied and one virtual orbital, at every step of the variational process. As an example, the potential energy surfaces of the S0 ground and 1S1(n, π*) first excited state of the formic acid HCOOH are calculated.

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Symmetry-Adapted Perturbation with Half-Projection for Spin Unrestricted Geminals

Mihálka

Śurján

Szabados

2021

J. Chem. Theory Comput.

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Perturbative correction to a wave function built from singlet−triplet mixed two-electron functions (geminals) is derived in the context of symmetry-adapted schemes, applying partial spinprojection. Imposing the constraint of strong orthogonality of geminals results in a reference function that captures static correlation in a computationally feasible way. In case of a lack of spin purification, the product of spin-unrestricted geminals is generally spincontaminated, potentially undermining performance of a subsequent dynamic correlation treatment. In this work, spin symmetry of the reference is partially restored by half-projection in a variation-afterprojection scheme. Applying perturbation theory (PT) to recover the missing part of electron correlation is hampered by the fact that an obvious choice for a zero-order Hamiltonian is not provided. The situation is amended by adopting the formalism of symmetryadapted PT. The resulting framework is tested on singlet−triplet gaps of biradicaloids, and it is found to perform well in situations where its unprojected counterpart fails because of spin contamination.

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The practical implementation of Löwdin's method for spin projection

Cited by 7 publications

References 34 publications

A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

Half‐Projected Hartree–Fock method: History and application to excited states of the same symmetry as the ground state

Symmetry-Adapted Perturbation with Half-Projection for Spin Unrestricted Geminals

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