1
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Abstract
The computational modeling of fullerenes plays a fundamental role in designing low-dimension carbon nanostructures. Nevertheless, the relative energies of fullerenes larger than C20 and C24 have not been comprehensively examined by means of highly accurate ab initio methods, for example, the CCSD(T) method. Here we report such an investigation for a diverse set of 29 C40 isomers. We calculate the energies of the C40 fullerenes using the G4(MP2) composite ab initio method, which approximates the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). The CCSD(T)/TZ isomerization energies span 43.1-763.3 kJ mol-1. We find a linear correlation (R2 = 0.96) between the CCSD(T)/TZ isomerization energies and the fullerene pentagon signatures (P1 index), which reflect the strain associated with fused pentagon-pentagon rings. Using the reference CCSD(T)/TZ isomerization energies, we examine the relationship between the percentage of exact Hartree-Fock (HF) exchange in hybrid density functional theory (DFT) methods and the pentagon-pentagon strain energies. We find that the performance of hybrid DFT methods deteriorates with the pentagon-pentagon strain energy. This deterioration in performance becomes more pronounced with the inclusion of high amounts of HF exchange. For example, for B3LYP (20% HF exchange), the root-mean-square deviation (RMSD) relative to G4(MP2) increases from 8.9 kJ mol-1 for the low-strain isomers (P1 = 11) to 18.0 kJ mol-1 for the high-strain isomers (P1 > 13). However, for BH&HLYP (50% HF exchange) the RMSD increases from 23.0 (P1 = 11) to 113.2 (P1 > 13) kJ mol-1. A similar trend is observed for the M06/M06-2X pair of functionals. Namely, for M06 (27% HF exchange) the RMSD increases from 0.8 (P1 = 11) to 21.0 (P1 > 13) kJ mol-1, whereas for M06-2X (54% HF exchange) the RMSD increases from 16.7 (P1 = 11) to 77.7 (P1 > 13) kJ mol-1. Overall, we find that the strain associated with pentagon adjacency is an inherently challenging problem for hybrid DFT methods involving high amounts of HF exchange and that there is an inverse relationship between the optimal percentage of HF exchange and the pentagon-pentagon strain energy. For example, for BLYP the optimal percentages of HF exchange are 13% (P1 = 11), 10% (P1 = 12), 7.5% (P1 = 13), and 6% (P1 > 13).
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Affiliation(s)
- Amir Karton
- School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
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2
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Abstract
Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.
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Affiliation(s)
- Prachi Sharma
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA;
| | - Jie J Bao
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA;
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA;
| | - Laura Gagliardi
- Department of Chemistry, Pritzker School of Molecular Engineering, James Franck Institute, and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, Illinois 60637, USA;
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3
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Shu Y, Truhlar DG. Relationships between Orbital Energies, Optical and Fundamental Gaps, and Exciton Shifts in Approximate Density Functional Theory and Quasiparticle Theory. J Chem Theory Comput 2020; 16:4337-4350. [PMID: 32453951 DOI: 10.1021/acs.jctc.0c00320] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Abstract
The relationships between Kohn-Sham (KS) and generalized KS (GKS) density functional orbital energies and fundamental gaps or optical gaps raise many interesting questions including the physical meanings of KS and GKS orbital energies when computed with currently available approximate density functionals (ADFs). In this work, by examining three diverse databases with various ADFs, we examine such relations from the point of view of the exciton shift of quasiparticle theory. We start by calculating a large number of excitation energies by time-dependent density functional theory (TDDFT) with a large number of ADFs. To relate the exciton shift implicit in TDDFT to the exciton shift that is explicit in Green's function theory, we define the exciton shift in TDDFT as the difference of the response shift and the quasiparticle shift. We found a strong correlation between the response shift and the amount of Hartree-Fock exchange included in the density functional, with the response shift varying between -1 and 5 eV. This range is an order of magnitude larger than the mean errors of the TDDFT excitation energies. This result suggests that, with currently available functionals, the KS or GKS orbital energies should be treated as intermediate mathematical variables in the calculation of excitation energies rather than as the energies of independent-particle reference states for quasiparticle theory.
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Affiliation(s)
- Yinan Shu
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States
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4
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Mostafanejad M, Liebenthal MD, DePrince AE. Global Hybrid Multiconfiguration Pair-Density Functional Theory. J Chem Theory Comput 2020; 16:2274-2283. [PMID: 32101416 DOI: 10.1021/acs.jctc.9b01178] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A global hybrid extension of multiconfiguration pair-density functional theory (MC-PDFT) is developed. Using a linear decomposition of the electron-electron repulsion term, a fraction λ of the nonlocal exchange interaction, obtained from variational two-electron reduced-density matrix (v2RDM)-driven complete active-space self-consistent field (CASSCF) theory, is combined with its local counterpart, obtained from an on-top pair-density functional. The resulting scheme (called λ-MC-PDFT) inherits the benefits of MC-PDFT (e.g., its simplicity and the resolution of the symmetry dilemma) and, when combined with the v2RDM approach to CASSCF, requires only polynomially scaling computational effort. As a result, λ-MC-PDFT can efficiently describe static and dynamical correlation effects in strongly correlated systems. The efficacy of the approach is assessed for several challenging multiconfigurational problems, including the dissociation of molecular nitrogen, the double dissociation of a water molecule, and the 1,3-dipolar cycloadditions of ozone to ethylene and ozone to acetylene in the O3ADD6 benchmark set.
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Affiliation(s)
- Mohammad Mostafanejad
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States
| | - Marcus Dante Liebenthal
- Department of Chemistry and Biochemistry, Ithaca College, Ithaca, New York 14850, United States
| | - A Eugene DePrince
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States
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5
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Mostafanejad M, Haney J, DePrince AE. Kinetic-energy-based error quantification in Kohn–Sham density functional theory. Phys Chem Chem Phys 2019; 21:26492-26501. [DOI: 10.1039/c9cp04595c] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present a basis-independent metric to assess the quality of the electron density obtained from Kohn–Sham (KS) density functional theory (DFT).
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Affiliation(s)
| | - Jessica Haney
- Department of Chemistry and Biochemistry
- Florida State University
- Tallahassee
- USA
| | - A. Eugene DePrince
- Department of Chemistry and Biochemistry
- Florida State University
- Tallahassee
- USA
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6
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Mostafanejad M, DePrince AE. Combining Pair-Density Functional Theory and Variational Two-Electron Reduced-Density Matrix Methods. J Chem Theory Comput 2018; 15:290-302. [DOI: 10.1021/acs.jctc.8b00988] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Mohammad Mostafanejad
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States
| | - A. Eugene DePrince
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States
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7
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Verma P, Truhlar DG. HLE16: A Local Kohn-Sham Gradient Approximation with Good Performance for Semiconductor Band Gaps and Molecular Excitation Energies. J Phys Chem Lett 2017; 8:380-387. [PMID: 28033712 DOI: 10.1021/acs.jpclett.6b02757] [Citation(s) in RCA: 43] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
Local exchange-correlation functionals have low cost and convenient portability but are known to seriously underestimate semiconductor band gaps and the energies of molecular Rydberg states. Here we present a new local approximation to the exchange-correlation functional called HLE16 that gives good performance for semiconductor band gaps and molecular excitation energies and is competitive with hybrid functionals. By the simultaneous increase of the local exchange and decrease of the local correlation, electronic excitation energies were improved without excessively degrading the ground-state solid-state cohesive energies, molecular bond energies, or chemical reaction barrier heights, although the new functional is not recommended for optimizing lattice constants or molecular bond lengths. The new functional can be useful as-is for calculations on semiconductors or excited states where it is essential to control the cost, and it can also be useful in establishing a starting point for developing even better new functionals that perform well for excited states.
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Affiliation(s)
- Pragya Verma
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota , Minneapolis, Minnesota 55455-0431, United States
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota , Minneapolis, Minnesota 55455-0431, United States
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8
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Verma P, Varga Z, Klein JEMN, Cramer CJ, Que L, Truhlar DG. Assessment of electronic structure methods for the determination of the ground spin states of Fe(ii), Fe(iii) and Fe(iv) complexes. Phys Chem Chem Phys 2017; 19:13049-13069. [DOI: 10.1039/c7cp01263b] [Citation(s) in RCA: 69] [Impact Index Per Article: 9.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We studied spin states of Fe2+ ion, gaseous FeO, and 14 Fe(ii), Fe(iii) and Fe(iv) complexes using density functional theory.
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Affiliation(s)
- Pragya Verma
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Chemical Theory Center and Minnesota Supercomputing Institute
| | - Zoltan Varga
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Chemical Theory Center and Minnesota Supercomputing Institute
| | - Johannes E. M. N. Klein
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Center for Metals in Biocatalysis
| | - Christopher J. Cramer
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Chemical Theory Center and Minnesota Supercomputing Institute
| | - Lawrence Que
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Center for Metals in Biocatalysis
| | - Donald G. Truhlar
- Department of Chemistry
- University of Minnesota
- Minneapolis
- USA
- Chemical Theory Center and Minnesota Supercomputing Institute
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9
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Yu HS, He X, Li SL, Truhlar DG. MN15: A Kohn-Sham global-hybrid exchange-correlation density functional with broad accuracy for multi-reference and single-reference systems and noncovalent interactions. Chem Sci 2016; 7:5032-5051. [PMID: 30155154 PMCID: PMC6018516 DOI: 10.1039/c6sc00705h] [Citation(s) in RCA: 701] [Impact Index Per Article: 87.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2016] [Accepted: 04/06/2016] [Indexed: 01/12/2023] Open
Abstract
Kohn-Sham density functionals are widely used; however, no currently available exchange-correlation functional can predict all chemical properties with chemical accuracy. Here we report a new functional, called MN15, that has broader accuracy than any previously available one. The properties considered in the parameterization include bond energies, atomization energies, ionization potentials, electron affinities, proton affinities, reaction barrier heights, noncovalent interactions, hydrocarbon thermochemistry, isomerization energies, electronic excitation energies, absolute atomic energies, and molecular structures. When compared with 82 other density functionals that have been defined in the literature, MN15 gives the second smallest mean unsigned error (MUE) for 54 data on inherently multiconfigurational systems, the smallest MUE for 313 single-reference chemical data, and the smallest MUE on 87 noncovalent data, with MUEs for these three categories of 4.75, 1.85, and 0.25 kcal mol-1, respectively, as compared to the average MUEs of the other 82 functionals of 14.0, 4.63, and 1.98 kcal mol-1. The MUE for 17 absolute atomic energies is 7.4 kcal mol-1 as compared to an average MUE of the other 82 functionals of 34.6 kcal mol-1. We further tested MN15 for 10 transition-metal coordination energies, the entire S66x8 database of noncovalent interactions, 21 transition-metal reaction barrier heights, 69 electronic excitation energies of organic molecules, 31 semiconductor band gaps, seven transition-metal dimer bond lengths, and 193 bond lengths of 47 organic molecules. The MN15 functional not only performs very well for our training set, which has 481 pieces of data, but also performs very well for our test set, which has 823 data that are not in our training set. The test set includes both ground-state properties and molecular excitation energies. For the latter MN15 achieves simultaneous accuracy for both valence and Rydberg electronic excitations when used with linear-response time-dependent density functional theory, with an MUE of less than 0.3 eV for both types of excitations.
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Affiliation(s)
- Haoyu S Yu
- Department of Chemistry , Chemical Theory Center , Inorganometallic Catalyst Design Center , Minnesota Supercomputing Institute , University of Minnesota , Minneapolis , Minnesota 55455-0431 , USA .
| | - Xiao He
- Department of Chemistry , Chemical Theory Center , Inorganometallic Catalyst Design Center , Minnesota Supercomputing Institute , University of Minnesota , Minneapolis , Minnesota 55455-0431 , USA .
- State Key Laboratory of Precision Spectroscopy and Department of Physics , East China Normal University , Shanghai , 200062 , China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai , Shanghai , 200062 , China
| | - Shaohong L Li
- Department of Chemistry , Chemical Theory Center , Inorganometallic Catalyst Design Center , Minnesota Supercomputing Institute , University of Minnesota , Minneapolis , Minnesota 55455-0431 , USA .
| | - Donald G Truhlar
- Department of Chemistry , Chemical Theory Center , Inorganometallic Catalyst Design Center , Minnesota Supercomputing Institute , University of Minnesota , Minneapolis , Minnesota 55455-0431 , USA .
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10
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Li X, Xu X, You X, Truhlar DG. Benchmark Calculations for Bond Dissociation Enthalpies of Unsaturated Methyl Esters and the Bond Dissociation Enthalpies of Methyl Linolenate. J Phys Chem A 2016; 120:4025-36. [DOI: 10.1021/acs.jpca.6b02600] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Xiaoyu Li
- Center
for Combustion Energy, Tsinghua University, Beijing 100084, China
- Department
of Chemistry, Chemical Theory Center, and Minnesota Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Xuefei Xu
- Center
for Combustion Energy, Tsinghua University, Beijing 100084, China
- Department
of Chemistry, Chemical Theory Center, and Minnesota Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Xiaoqing You
- Center
for Combustion Energy, Tsinghua University, Beijing 100084, China
| | - Donald G. Truhlar
- Department
of Chemistry, Chemical Theory Center, and Minnesota Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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11
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Yu HS, He X, Truhlar DG. MN15-L: A New Local Exchange-Correlation Functional for Kohn-Sham Density Functional Theory with Broad Accuracy for Atoms, Molecules, and Solids. J Chem Theory Comput 2016; 12:1280-93. [PMID: 26722866 DOI: 10.1021/acs.jctc.5b01082] [Citation(s) in RCA: 300] [Impact Index Per Article: 37.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Kohn-Sham density functional theory is widely used for applications of electronic structure theory in chemistry, materials science, and condensed-matter physics, but the accuracy depends on the quality of the exchange-correlation functional. Here, we present a new local exchange-correlation functional called MN15-L that predicts accurate results for a broad range of molecular and solid-state properties including main-group bond energies, transition metal bond energies, reaction barrier heights, noncovalent interactions, atomic excitation energies, ionization potentials, electron affinities, total atomic energies, hydrocarbon thermochemistry, and lattice constants of solids. The MN15-L functional has the same mathematical form as a previous meta-nonseparable gradient approximation exchange-correlation functional, MN12-L, but it is improved because we optimized it against a larger database, designated 2015A, and included smoothness restraints; the optimization has a much better representation of transition metals. The mean unsigned error on 422 chemical energies is 2.32 kcal/mol, which is the best among all tested functionals, with or without nonlocal exchange. The MN15-L functional also provides good results for test sets that are outside the training set. A key issue is that the functional is local (no nonlocal exchange or nonlocal correlation), which makes it relatively economical for treating large and complex systems and solids. Another key advantage is that medium-range correlation energy is built in so that one does not need to add damped dispersion by molecular mechanics in order to predict accurate noncovalent binding energies. We believe that the MN15-L functional should be useful for a wide variety of applications in chemistry, physics, materials science, and molecular biology.
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Affiliation(s)
- Haoyu S Yu
- Department of Chemistry, Chemical Theory Center, Inorganometallic Catalyst Design Center, and Supercomputing Institute, University of Minnesota , Minneapolis, Minnesota 55455-0431, United States
| | - Xiao He
- Department of Chemistry, Chemical Theory Center, Inorganometallic Catalyst Design Center, and Supercomputing Institute, University of Minnesota , Minneapolis, Minnesota 55455-0431, United States.,State Key Laboratory of Precision Spectroscopy, East China Normal University , Shanghai 200062, China
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, Inorganometallic Catalyst Design Center, and Supercomputing Institute, University of Minnesota , Minneapolis, Minnesota 55455-0431, United States
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12
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Li Manni G, Carlson RK, Luo S, Ma D, Olsen J, Truhlar DG, Gagliardi L. Multiconfiguration Pair-Density Functional Theory. J Chem Theory Comput 2015; 10:3669-80. [PMID: 26588512 DOI: 10.1021/ct500483t] [Citation(s) in RCA: 299] [Impact Index Per Article: 33.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We present a new theoretical framework, called Multiconfiguration Pair-Density Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of density functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic density, its gradient, the on-top pair density, and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total density, its gradient, and the on-top pair density to calculate the remaining part of the energy, which we call the on-top-density-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair density is an element of the two-particle density matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle density. To illustrate the theory, we obtain first approximations to the required new type of density functionals by translating conventional density functionals of the spin densities using a simple prescription, and we perform post-SCF density functional calculations using the total density, density gradient, and on-top pair density from the MCSCF calculations. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N(+), O, O(+), Sc(+), Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quantitative accuracy, even with the present first approximations to the new types of density functionals, that is comparable to much more expensive multireference perturbation theory methods.
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Affiliation(s)
- Giovanni Li Manni
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
| | - Rebecca K Carlson
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
| | - Sijie Luo
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
| | - Dongxia Ma
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
| | - Jeppe Olsen
- Department of Chemistry, Aarhus University , Langelandsgade 140, DK-8000, Aarhus C, Denmark
| | - Donald G Truhlar
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
| | - Laura Gagliardi
- Department of Chemistry, Supercomputing Institute, and Chemical Theory Center, University of Minnesota , Minneapolis, Minnesota 55455, United States
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13
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Yu H, Truhlar DG. Components of the Bond Energy in Polar Diatomic Molecules, Radicals, and Ions Formed by Group-1 and Group-2 Metal Atoms. J Chem Theory Comput 2015; 11:2968-83. [DOI: 10.1021/acs.jctc.5b00083] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Haoyu Yu
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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14
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Li SL, Truhlar DG. Improving Rydberg Excitations within Time-Dependent Density Functional Theory with Generalized Gradient Approximations: The Exchange-Enhancement-for-Large-Gradient Scheme. J Chem Theory Comput 2015; 11:3123-30. [DOI: 10.1021/acs.jctc.5b00369] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Shaohong L. Li
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Donald G. Truhlar
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
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15
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Xu X, Zhang W, Tang M, Truhlar DG. Do Practical Standard Coupled Cluster Calculations Agree Better than Kohn–Sham Calculations with Currently Available Functionals When Compared to the Best Available Experimental Data for Dissociation Energies of Bonds to 3d Transition Metals? J Chem Theory Comput 2015; 11:2036-52. [DOI: 10.1021/acs.jctc.5b00081] [Citation(s) in RCA: 94] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Xuefei Xu
- Department
of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Wenjing Zhang
- Department
of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
- College
of Chemistry and Molecular Engineering, Zhengzhou University, Zhengzhou, Henan Province 450001, People’s Republic of China
| | - Mingsheng Tang
- College
of Chemistry and Molecular Engineering, Zhengzhou University, Zhengzhou, Henan Province 450001, People’s Republic of China
| | - Donald G. Truhlar
- Department
of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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16
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Yu HS, Zhang W, Verma P, He X, Truhlar DG. Nonseparable exchange–correlation functional for molecules, including homogeneous catalysis involving transition metals. Phys Chem Chem Phys 2015; 17:12146-60. [DOI: 10.1039/c5cp01425e] [Citation(s) in RCA: 96] [Impact Index Per Article: 10.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022]
Abstract
A gradient approximation, GAM, to the exchange–correlation functional of Kohn–Sham theory with broad performance for metal and nonmetal bond energies and weak interactions is reported.
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Affiliation(s)
- Haoyu S. Yu
- Department of Chemistry
- Chemical Theory Center
- Inorganometallic Catalyst Design Center
- and Supercomputing Institute
- University of Minnesota
| | - Wenjing Zhang
- Department of Chemistry
- Chemical Theory Center
- Inorganometallic Catalyst Design Center
- and Supercomputing Institute
- University of Minnesota
| | - Pragya Verma
- Department of Chemistry
- Chemical Theory Center
- Inorganometallic Catalyst Design Center
- and Supercomputing Institute
- University of Minnesota
| | - Xiao He
- Department of Chemistry
- Chemical Theory Center
- Inorganometallic Catalyst Design Center
- and Supercomputing Institute
- University of Minnesota
| | - Donald G. Truhlar
- Department of Chemistry
- Chemical Theory Center
- Inorganometallic Catalyst Design Center
- and Supercomputing Institute
- University of Minnesota
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17
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Li SL, Truhlar DG. Testing time-dependent density functional theory with depopulated molecular orbitals for predicting electronic excitation energies of valence, Rydberg, and charge-transfer states and potential energies near a conical intersection. J Chem Phys 2014; 141:104106. [DOI: 10.1063/1.4894522] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Shaohong L. Li
- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
| | - Donald G. Truhlar
- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
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18
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Zhang W, Truhlar DG, Tang M. Explanation of the Source of Very Large Errors in Many Exchange–Correlation Functionals for Vanadium Dimer. J Chem Theory Comput 2014; 10:2399-409. [DOI: 10.1021/ct500296a] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Wenjing Zhang
- The
College of Chemistry and Molecular Engineering, Zhengzhou University, Zhengzhou, Henan Province 450001, China
- Department
of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department
of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Mingsheng Tang
- The
College of Chemistry and Molecular Engineering, Zhengzhou University, Zhengzhou, Henan Province 450001, China
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19
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Yu H, Truhlar DG. What Dominates the Error in the CaO Diatomic Bond Energy Predicted by Various Approximate Exchange–Correlation Functionals? J Chem Theory Comput 2014; 10:2291-305. [DOI: 10.1021/ct5000814] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Haoyu Yu
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry,
Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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20
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Xu X, Yang KR, Truhlar DG. Testing Noncollinear Spin-Flip, Collinear Spin-Flip, and Conventional Time-Dependent Density Functional Theory for Predicting Electronic Excitation Energies of Closed-Shell Atoms. J Chem Theory Comput 2014; 10:2070-84. [DOI: 10.1021/ct500128s] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
Affiliation(s)
- Xuefei Xu
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Ke R. Yang
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Donald G. Truhlar
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
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21
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Peverati R, Truhlar DG. Quest for a universal density functional: the accuracy of density functionals across a broad spectrum of databases in chemistry and physics. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:20120476. [PMID: 24516178 DOI: 10.1098/rsta.2012.0476] [Citation(s) in RCA: 467] [Impact Index Per Article: 46.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
Kohn-Sham density functional theory is in principle an exact formulation of quantum mechanical electronic structure theory, but in practice we have to rely on approximate exchange-correlation (xc) functionals. The objective of our work has been to design an xc functional with broad accuracy across as wide an expanse of chemistry and physics as possible, leading--as a long-range goal--to a functional with good accuracy for all problems, i.e. a universal functional. To guide our path towards that goal and to measure our progress, we have developed-building on earlier work of our group-a set of databases of reference data for a variety of energetic and structural properties in chemistry and physics. These databases include energies of molecular processes, such as atomization, complexation, proton addition and ionization; they also include molecular geometries and solid-state lattice constants, chemical reaction barrier heights, and cohesive energies and band gaps of solids. For this paper, we gather many of these databases into four comprehensive databases, two with 384 energetic data for chemistry and solid-state physics and another two with 68 structural data for chemistry and solid-state physics, and we test two wave function methods and 77 density functionals (12 Minnesota meta functionals and 65 others) in a consistent way across this same broad set of data. We especially highlight the Minnesota density functionals, but the results have broader implications in that one may see the successes and failures of many kinds of density functionals when they are all applied to the same data. Therefore, the results provide a status report on the quest for a universal functional.
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Affiliation(s)
- Roberto Peverati
- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, , Minneapolis, MN 55455-0431, USA
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22
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Luo S, Averkiev B, Yang KR, Xu X, Truhlar DG. Density Functional Theory of Open-Shell Systems. The 3d-Series Transition-Metal Atoms and Their Cations. J Chem Theory Comput 2013; 10:102-21. [DOI: 10.1021/ct400712k] [Citation(s) in RCA: 53] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Sijie Luo
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Boris Averkiev
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Ke R. Yang
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Xuefei Xu
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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23
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Mardirossian N, Head-Gordon M. Characterizing and Understanding the Remarkably Slow Basis Set Convergence of Several Minnesota Density Functionals for Intermolecular Interaction Energies. J Chem Theory Comput 2013; 9:4453-61. [PMID: 26589163 DOI: 10.1021/ct400660j] [Citation(s) in RCA: 72] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
For a set of eight equilibrium intermolecular complexes, it is discovered that the basis set limit (BSL) cannot be reached by aug-cc-pV5Z for three of the Minnesota density functionals: M06-L, M06-HF, and M11-L. In addition, the M06 and M11 functionals exhibit substantial, but less severe, difficulties in reaching the BSL. By using successively finer grids, it is demonstrated that this issue is not related to the numerical integration of the exchange-correlation functional. In addition, it is shown that the difficulty in reaching the BSL is not a direct consequence of the structure of the augmented functions in Dunning's basis sets, since modified augmentation yields similar results. By using a very large custom basis set, the BSL appears to be reached for the HF dimer for all of the functionals. As a result, it is concluded that the difficulties faced by several of the Minnesota density functionals are related to an interplay between the form of these functionals and the structure of standard basis sets. It is speculated that the difficulty in reaching the basis set limit is related to the magnitude of the inhomogeneity correction factor (ICF) of the exchange functional. A simple modification of the M06-L exchange functional that systematically reduces the basis set superposition error (BSSE) for the HF dimer in the aug-cc-pVQZ basis set is presented, further supporting the speculation that the difficulty in reaching the BSL is caused by the magnitude of the exchange functional ICF. Finally, the BSSE is plotted with respect to the internuclear distance of the neon dimer for two of the examined functionals.
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Affiliation(s)
- Narbe Mardirossian
- Department of Chemistry, University of California , Berkeley, California 94720, United States
| | - Martin Head-Gordon
- Chemical Sciences Division, Lawrence Berkeley National Laboratory , Berkeley, California 94720, United States
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24
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Zhang W, Truhlar DG, Tang M. Tests of Exchange-Correlation Functional Approximations Against Reliable Experimental Data for Average Bond Energies of 3d Transition Metal Compounds. J Chem Theory Comput 2013; 9:3965-77. [DOI: 10.1021/ct400418u] [Citation(s) in RCA: 82] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Affiliation(s)
- Wenjing Zhang
- College of Chemistry and Molecular
Engineering, Zhengzhou University, Zhengzhou,
Henan Province 450001, P. R. China
- Department of Chemistry and Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry and Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Mingsheng Tang
- College of Chemistry and Molecular
Engineering, Zhengzhou University, Zhengzhou,
Henan Province 450001, P. R. China
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25
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Peverati R, Truhlar DG. Screened-exchange density functionals with broad accuracy for chemistry and solid-state physics. Phys Chem Chem Phys 2012; 14:16187-91. [PMID: 23132141 DOI: 10.1039/c2cp42576a] [Citation(s) in RCA: 404] [Impact Index Per Article: 33.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present two new exchange-correlation functionals for hybrid Kohn-Sham electronic structure calculations based on the nonseparable functional form introduced recently in the N12 and MN12-L functionals but now with the addition of screened Hartree-Fock exchange. The first functional depends on the density and the density gradient and is called N12-SX; the second functional depends on the density, the density gradient, and the kinetic energy density and is called MN12-SX. Both new functionals include a portion of the Hartree-Fock exchange at short-range, but Hartree-Fock exchange is screened at long range. The accuracies of the two new functionals are compared to those of the recent N12 and MN12-L local functionals to show the effect of adding screened exchange, are compared to the previously best available screened exchange functional, HSE06, and are compared to the best available global-hybrid generalized gradient approximation (GGA) and to a high-performance long-range-corrected meta-GGA.
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Affiliation(s)
- Roberto Peverati
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, USA
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26
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Luo S, Truhlar DG. How Evenly Can Approximate Density Functionals Treat the Different Multiplicities and Ionization States of 4d Transition Metal Atoms? J Chem Theory Comput 2012; 8:4112-26. [DOI: 10.1021/ct300737t] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Sijie Luo
- Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431,
United States
| | - Donald G. Truhlar
- Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431,
United States
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27
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Zhao Y, Ng HT, Peverati R, Truhlar DG. Benchmark Database for Ylidic Bond Dissociation Energies and Its Use for Assessments of Electronic Structure Methods. J Chem Theory Comput 2012; 8:2824-34. [DOI: 10.1021/ct300457c] [Citation(s) in RCA: 56] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Yan Zhao
- Printing Content & Delivery Laboratory, HP Laboratories, Hewlett-Packard Company, 1501 Page Mill Road, Palo Alto, California 94304, United States
| | - Hou T. Ng
- Printing Content & Delivery Laboratory, HP Laboratories, Hewlett-Packard Company, 1501 Page Mill Road, Palo Alto, California 94304, United States
| | - Roberto Peverati
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry and
Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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28
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Peverati R, Truhlar DG. Exchange–Correlation Functional with Good Accuracy for Both Structural and Energetic Properties while Depending Only on the Density and Its Gradient. J Chem Theory Comput 2012; 8:2310-9. [DOI: 10.1021/ct3002656] [Citation(s) in RCA: 235] [Impact Index Per Article: 19.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Roberto Peverati
- Department of Chemistry and Supercomputing
Institute,
University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry and Supercomputing
Institute,
University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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29
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Fliszár S, Vauthier ÉC. On quasi-transferable molecular fragments. Part V: Free radicals; the dissociation of organic molecules, an improved Xα(LSD) approach. COMPUT THEOR CHEM 2011. [DOI: 10.1016/j.comptc.2011.04.022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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30
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Affiliation(s)
- A. Nagy
- a Institute of Theoretical Physics, Kossuth Lajos University , H-4010 , Debrecen , Hungary
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31
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Garza J, Vargas R, Nichols JA, Dixon DA. Orbital energy analysis with respect to LDA and self-interaction corrected exchange-only potentials. J Chem Phys 2001. [DOI: 10.1063/1.1327269] [Citation(s) in RCA: 61] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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32
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Chattaraj P, Sengupta S, Poddar A. Quantum fluid density functional theory of chemical reactivity in a two-state ensemble. ACTA ACUST UNITED AC 2000. [DOI: 10.1016/s0166-1280(99)00444-3] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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33
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On the resolution of identity Coulomb energy approximation in density functional theory. ACTA ACUST UNITED AC 2000. [DOI: 10.1016/s0166-1280(99)00434-0] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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34
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Liu S, Ayers PW, Parr RG. Alternative definition of exchange-correlation charge in density functional theory. J Chem Phys 1999. [DOI: 10.1063/1.479924] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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35
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Gaspar R, Nagy A. X alpha method with theoretically determined parameter α- exchange parameters α and ionisation energies of multiply charged ions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/20/15/012] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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36
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Andrejkovics I, Nagy Á. Excitation energies in density functional theory: comparison of several methods for the H2O, N2, CO and C2H4 molecules. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(98)01075-6] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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37
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Density functional theory for excited states. ADVANCES IN QUANTUM CHEMISTRY 1998. [DOI: 10.1016/s0065-3276(08)60268-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register]
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38
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Vauthier E, Blain M, Odiot S, Barone V, Comeau M, Fliszár S. Xα local spin density energy calculations. ACTA ACUST UNITED AC 1995. [DOI: 10.1016/0166-1280(95)04176-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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39
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Nagy Á. Relativistic density-functional theory for ensembles of excited states. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 49:3074-3076. [PMID: 9910590 DOI: 10.1103/physreva.49.3074] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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40
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Pathak RK, Kshirsagar A, Hoffmeyer R, Thakkar AJ. Leading corrections to atomic impulse-approximation Compton profiles: A density-functional approach. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1993; 48:2946-2951. [PMID: 9909943 DOI: 10.1103/physreva.48.2946] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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41
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Fliszár S, Desmarais N, Comeau M. Xα local spin density calculations. First-row diatomic molecules. ACTA ACUST UNITED AC 1991. [DOI: 10.1016/0166-1280(91)85135-t] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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42
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Bounds to Atomic and Molecular Energy Functionals. ADVANCES IN QUANTUM CHEMISTRY 1991. [DOI: 10.1016/s0065-3276(08)60365-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/14/2023]
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43
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Nagy Á. Parameter-free exchange potential for excitation in the density-functional theory: Application to excitation energies within the fractional-occupation approach. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:4388-4390. [PMID: 9904540 DOI: 10.1103/physreva.42.4388] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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44
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Gáspár R, Nagy Á. The first ionization energy, electron affinity and electronegativity calculated by theX α method with ab initio self-consistent exchange parameter. ACTA ACUST UNITED AC 1988. [DOI: 10.1007/bf03053845] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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45
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46
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Comparison of the orbitals of Neon, Argon and Krypton calculated by the hartree-fock and the Xα methods with several values of α. ACTA ACUST UNITED AC 1982. [DOI: 10.1007/bf03156194] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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