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Density functional theory with fractional orbital occupations
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10.1063/1.3703894
/content/aip/journal/jcp/136/15/10.1063/1.3703894
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/15/10.1063/1.3703894

Figures

Image of FIG. 1.
FIG. 1.

Potential energy curves (in total energy) for the ground state of H2, calculated by both the spin-restricted and spin-unrestricted formalisms of the HF theory and KS-DFT (with LDA and B3LYP functionals). The exact potential energy curve is calculated by the CCSD theory.

Image of FIG. 2.
FIG. 2.

Potential energy curves (in relative energy) for the ground state of H2, calculated by the spin-restricted HF theory and KS-DFT (with various XC functionals). The exact potential energy curve is calculated by the CCSD theory. The zeros of energy are set at the respective spin-unrestricted dissociation limits.

Image of FIG. 3.
FIG. 3.

Potential energy curves (in total energy) for the ground state of H2, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA.

Image of FIG. 4.
FIG. 4.

Same as Fig. 3 but in relative energy. The zeros of energy are set at the respective spin-unrestricted dissociation limits.

Image of FIG. 5.
FIG. 5.

Occupation numbers of the 1σ g orbital for the ground state of H2 as a function of the internuclear distance R, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA. The reference data are the FCI NOONs.37

Image of FIG. 6.
FIG. 6.

Potential energy curves (in relative energy) for the ground state of H2, calculated by the spin-restricted (with and without the entropy contributions) and spin-unrestricted TAO-LDA (θ = 40 mHartree), where the zeros of energy are set at the spin-unrestricted dissociation limit. The entropy contributions (in total energy) as a function of the internuclear distance R, calculated by spin-restricted TAO-LDA (θ = 40 mHartree), are also shown.

Image of FIG. 7.
FIG. 7.

Same as Fig. 6 but for θ = 7 mHartree.

Image of FIG. 8.
FIG. 8.

Potential energy curves (in total energy) for the ground state of N2, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA.

Image of FIG. 9.
FIG. 9.

Same as Fig. 8 but in relative energy. The zeros of energy are set at the respective spin-unrestricted dissociation limits.

Image of FIG. 10.
FIG. 10.

Occupation numbers of the 3σ g orbital for the ground state of N2 as a function of the internuclear distance R, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA. The reference data are the MRCI NOONs.77

Image of FIG. 11.
FIG. 11.

Same as Fig. 10 but for the 1π ux orbital.

Image of FIG. 12.
FIG. 12.

Potential energy curves (in relative energy) for the ground state of N2, calculated by the spin-restricted (with and without the entropy contributions) and spin-unrestricted TAO-LDA (θ = 40 mHartree), where the zeros of energy are set at the spin-unrestricted dissociation limit. The entropy contributions (in total energy) as a function of the internuclear distance R, calculated by spin-restricted TAO-LDA (θ = 40 mHartree), are also shown.

Image of FIG. 13.
FIG. 13.

Torsion potential energy curves (in relative energy) for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated by spin-restricted TAO-LDA (with various θ). The zeros of energy are set at the respective minimum energies. The θ = 0 case corresponds to KS-LDA.

Image of FIG. 14.
FIG. 14.

Occupation numbers of the π (1b2) orbital for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA. The reference data are the half-projected NOONs of CASSCF method (HPNO-CAS).79

Image of FIG. 15.
FIG. 15.

Torsion potential energy curves (in relative energy) for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated by the spin-restricted (with and without the entropy contributions) and spin-unrestricted TAO-LDA (θ = 15 mHartree), where the zeros of energy are set at the respective minimum energies. The entropy contributions (in total energy) as a function of the HCCH torsion angle, calculated by spin-restricted TAO-LDA (θ = 15 mHartree), are also shown.

Image of FIG. 16.
FIG. 16.

Pentacene, consisting of 5 linearly fuzed benzene rings, is designated as 5-acene.

Image of FIG. 17.
FIG. 17.

Singlet-triplet energy gap as a function of the acene length, calculated by spin-unrestricted KS-DFT (with LDA, BLYP, and B3LYP functionals), using the 6-31G* basis set. The experimental data are taken from Refs. 80 to 83, and the DMRG data are taken from Ref. 88.

Image of FIG. 18.
FIG. 18.

Singlet-triplet energy gap as a function of the acene length, calculated by spin-unrestricted TAO-LDA (with various θ), using the 6-31G* basis set. The θ = 0 case corresponds to spin-unrestricted KS-LDA. The experimental data are taken from Refs. 80 to 83.

Image of FIG. 19.
FIG. 19.

Singlet-triplet energy gap as a function of the acene length, calculated by spin-unrestricted TAO-LDA (θ = 7 mHartree), using both the 6-31G* and 6-31G basis sets.

Image of FIG. 20.
FIG. 20.

HOMO occupation numbers for the lowest singlet states of n-acenes as a function of the acene length, calculated by spin-restricted TAO-LDA (with various θ)/6-31G*. The θ = 0 case corresponds to spin-restricted KS-LDA. Reference data are the NOONs of the active-space variational 2-RDM method.95

Image of FIG. 21.
FIG. 21.

Active orbital occupation numbers (HOMO-6, …, HOMO-1, HOMO, LUMO, LUMO+1, …, and LUMO+6) for the lowest singlet states of n-acenes as a function of the acene length, calculated by spin-restricted TAO-LDA (θ = 7 mHartree)/6-31G*.

Tables

Generic image for table
Table I.

Statistical errors (in kcal/mol) of the reaction energies of 30 chemical reactions, Ref. 20, calculated by TAO-LDA (with various θ (in mHartree)). The θ = 0 case corresponds to KS-LDA.

Generic image for table
Table II.

Statistical errors (in Å) of EXTS,74 calculated by TAO-LDA (with various θ (in mHartree)). The θ = 0 case corresponds to KS-LDA.

Generic image for table
Table III.

Singlet-triplet energy gaps (ST gaps) of n-acenes in the polymer limit (n → ∞), obtained by nonlinear least-squares fittings of 3 different data sets (20- to 74-acene, 30- to 74-acene, and 40- to 74-acene) of the ST gaps calculated by spin-unrestricted TAO-LDA (θ = 7 mHartree)/6-31G, using a power-law fitting function of the form a + bn c . Here, the coefficient of determination R 2 is a statistical measure of the goodness-of-fit (R 2 = 1, for a perfect fit).

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2012-04-17
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Density functional theory with fractional orbital occupations
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/15/10.1063/1.3703894
10.1063/1.3703894
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