基礎的概念

Background

In Hartree-Fock theory, the energy has the form:

E_HF = V + ⟨hP⟩ + 1/2⟨PJ(P)⟩ – 1/2⟨PK(P)⟩

where the terms have the following meanings:

In the Kohn-Sham formulation of density functional theory [ Kohn65 W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev., 140 (1965) A1133-A38. DOI: ], the exact exchange (HF) for a single determinant is replaced by a more general expression, the exchange-correlation functional, which can include terms accounting for both the exchange and the electron correlation energies, the latter not being present in Hartree-Fock theory:

E_KS = V + ⟨hP⟩ + 1/2⟨PJ(P)⟩ + E_X[P] + E_C[P]

where E_X[P] is the exchange functional, and E_C[P] is the correlation functional.

Within the Kohn-Sham formulation, Hartree-Fock theory can be regarded as a special case of density functional theory, with E_X[P] given by the exchange integral -1/2<PK(P)> and E_C=0. The functionals normally used in density functional theory are integrals of some function of the density and possibly the density gradient:

E_X[P] = ∫f(ρ_α(r),ρ_β(r),∇ρ_α(r),∇ρ_β(r))dr

where the methods differ in which function f is used for E_X and which (if any) f is used for E_C. In addition to pure DFT methods, Gaussian supports hybrid methods in which the exchange functional is a linear combination of the Hartree-Fock exchange and a functional integral of the above form. Proposed functionals lead to integrals which cannot be evaluated in closed form and are solved by numerical quadrature.

Hybrid関数キーワード

Keywords: Hybrid Functionals

A number of hybrid functionals, which include a mixture of Hartree-Fock exchange with DFT exchange-correlation, are available via keywords:

Becke Three-Parameter Hybrid Functionals

These functionals have the form devised by Becke in 1993 [ Becke93a A. D. Becke, “Density-functional thermochemistry. III. The role of exact exchange,” J. Chem. Phys., 98 (1993) 5648-52. DOI: ]:

A*E_X^Slater+(1-A)*E_X^HF+B*ΔE_X^Becke+E_C^VWN+C*ΔE_C^non-local

where A, B, and C are the constants determined by Becke via fitting to the G1 molecule set.

There are several variations of this hybrid functional.

B3LYP uses the non-local correlation provided by the LYP expression, and VWN functional III for local correlation (not functional V). Note that since LYP includes both local and non-local terms, the correlation functional used is actually:

C*E_C^LYP+(1-C)*E_C^VWN

In other words, VWN is used to provide the excess local correlation required, since LYP contains a local term essentially equivalent to VWN.

B3P86 specifies the same functional with the non-local correlation provided by Perdew 86, and B3PW91 specifies this functional with the non-local correlation provided by Perdew/Wang 91.

O3LYP is a three-parameter functional similar to B3LYP:

A*E_X^LSD+(1-A)*E_X^HF+B*ΔE_X^OPTX+C*ΔE_C^LYP+(1-C)E_C^VWN

where A, B and C are as defined by Cohen and Handy in reference [ Cohen01 A. J. Cohen and N. C. Handy, “Dynamic correlation,” Mol. Phys., 99 (2001) 607-15. DOI: ].

Functionals Including Dispersion

• APFD requests the Austin-Frisch-Petersson functional with dispersion [ Austin12 A. Austin, G. Petersson, M. J. Frisch, F. J. Dobek, G. Scalmani, and K. Throssell, “A density functional with spherical atom dispersion terms,” J. Chem. Theory and Comput. 8 (2012) 4989. DOI: ], and APF requests the same functional without dispersion.
• The wB97XD functional uses a version of Grimme’s D2 dispersion model.

The standalone keyword EmpiricalDispersion also allows you to specify a dispersion scheme with various functionals.

Long-Range-Corrected Functionals

The non-Coulomb part of exchange functionals typically dies off too rapidly and becomes very inaccurate at large distances, making them unsuitable for modeling processes such as electron excitations to high orbitals. Various schemes have been devised to handle such cases. Gaussian 16 offers the following functionals which include long-range corrections:

• LC-wHPBE: Recommended version [ Henderson09 T. M. Henderson, A. F. Izmaylov, G. Scalmani, and G. E. Scuseria, “Can short-range hybrids describe long-range-dependent properties?,” J. Chem. Phys., 131 (2009) 044108. DOI: ] of the long-range-corrected ωPBE functional [ Vydrov06 O. A. Vydrov and G. E. Scuseria, “Assessment of a long range corrected hybrid functional,” J. Chem. Phys., 125 (2006) 234109. DOI: , Vydrov06a O. A. Vydrov, J. Heyd, A. Krukau, and G. E. Scuseria, “Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals,” J. Chem. Phys., 125 (2006) 074106. DOI: , Vydrov07 O. A. Vydrov, G. E. Scuseria, and J. P. Perdew, “Tests of functionals for systems with fractional electron number,” J. Chem. Phys., 126 (2007) 154109. DOI: ]. LC-wPBE requests the original version.
• CAM-B3LYP: Handy and coworkers’ long-range-corrected version of B3LYP using the Coulomb-attenuating method [ Yanai04 T. Yanai, D. Tew, and N. Handy, “A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP),” Chem. Phys. Lett., 393 (2004) 51-57. DOI: ].
• wB97XD: The latest functional from Head-Gordon and coworkers, which includes empirical dispersion [ Chai08a J.-D. Chai and M. Head-Gordon, “Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections,” Phys. Chem. Chem. Phys., 10 (2008) 6615-20. DOI: ]. The wB97 and wB97X [ Chai08 J.-D. Chai and M. Head-Gordon, “Systematic optimization of long-range corrected hybrid density functionals,” J. Chem. Phys., 128 (2008) 084106. DOI: ] variations are also available. These functionals also include long-range corrections.

In addition, the prefix LC- may be added to most pure functionals to apply the long correction of Hirao and coworkers [ Iikura01 H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, “Long-range correction scheme for generalized-gradient-approximation exchange functionals,” J. Chem. Phys., 115 (2001) 3540-44. DOI: ]: e.g., LC-BLYP.

Other Hybrid Functionals

Functionals from the Truhlar Group

• MN15 requests the MN15 [ Yu16 H. S. Yu, X. He, S. L. Li and D. G. Truhlar, “MN15: A Kohn-Sham Global-Hybrid Exchange-Correlation Density Functional with Broad Accuracy for Multi-Reference and Single-Reference Systems and Noncovalent Interactions,” Chemical Science 2016, 7, 5032-5051. DOI: ] functional.
• M11 [ Peverati11a R. Peverati and D. G. Truhlar, “Improving the Accuracy of Hybrid Meta-GGA Density Functionals by Range Separation,” J. Phys. Chem. Lett. 2 (2011) 2810-2817. DOI: ], SOGGA11X [ Peverati11b R. Peverati and D. G. Truhlar, “A global hybrid generalized gradient approximation to the exchange-correlation functional that satisfies the second-order density-gradient constraint and has broad applicability in chemistry,” J. Chem. Phys. 135 (2011) 191102. DOI: ], N12SX [ Peverati12a R. Peverati and D. G. Truhlar, “Screened-exchange density functionals with broad accuracy for chemistry and solidstate physics,” Phys. Chem. Chem. Phys. 14 (2012) 16187. DOI: ], and MN12SX [ Peverati12a R. Peverati and D. G. Truhlar, “Screened-exchange density functionals with broad accuracy for chemistry and solidstate physics,” Phys. Chem. Chem. Phys. 14 (2012) 16187. DOI: ] request these hybrid functionals from the Truhlar group.
• PW6B95 and PW6B95D3 [ Zhao05a Y. Zhao and D. G. Truhlar, “Design of Density Functionals That Are Broadly Accurate for Thermochemistry, Thermochemical Kinetics, and Nonbonded Interactions,” J. Phys. Chem. A, 2005, 109, 5656. DOI: ].
• M08HX: The M08-HX functional [ Zhao08a Y. Zhao and D. G. Truhlar, “Exploring the Limit of Accuracy of the Global Hybrid Meta Density Functional for Main-Group Thermochemistry, Kinetics, and Noncovalent Interactions,” J. Chem. Theory Compute. 2008, 4, 1849. DOI: ].
• M06 hybrid functional of Truhlar and Zhao [ Zhao08 Y. Zhao and D. G. Truhlar, “The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals,” Theor. Chem. Acc., 120 (2008) 215-41. DOI: ]. The M06HF [ Zhao06b Y. Zhao and D. G. Truhlar, “Comparative DFT study of van der Waals complexes: Rare-gas dimers, alkaline-earth dimers, zinc dimer, and zinc-rare-gas dimers,” J. Phys. Chem., 110 (2006) 5121-29. DOI: , Zhao06c Y. Zhao and D. G. Truhlar, “Density Functional for Spectroscopy: No Long-Range Self-Interaction Error, Good Performance for Rydberg and Charge-Transfer States, and Better Performance on Average than B3LYP for Ground States,” J. Phys. Chem. A, 110 (2006) 13126-30. DOI: ] and M062X [ Zhao08 Y. Zhao and D. G. Truhlar, “The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals,” Theor. Chem. Acc., 120 (2008) 215-41. DOI: ] variations are also available.
• M05 [ Zhao05 Y. Zhao, N. E. Schultz, and D. G. Truhlar, “Exchange-correlation functional with broad accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent interactions,” J. Chem. Phys., 123 (2005) 161103. DOI: ] and M052X [ Zhao06 Y. Zhao, N. E. Schultz, and D. G. Truhlar, “Design of density functionals by combining the method of constraint satisfaction with parametrization for thermochemistry, thermochemical kinetics, and noncovalent interactions,” J. Chem. Theory and Comput., 2 (2006) 364-82. DOI: ].

Functionals Employing PBE Correlation

• The 1996 pure functional of Perdew, Burke and Ernzerhof [ Perdew96a J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett., 77 (1996) 3865-68. DOI: , Perdew97 J. P. Perdew, K. Burke, and M. Ernzerhof, “Errata: Generalized gradient approximation made simple,” Phys. Rev. Lett., 78 (1997) 1396. DOI: ] as made into a hybrid functional by Adamo [ Adamo99a C. Adamo and V. Barone, “Toward reliable density functional methods without adjustable parameters: The PBE0 model,” J. Chem. Phys., 110 (1999) 6158-69. DOI: ]. The keyword is PBE1PBE. This functional uses 25% exact exchange and 75% DFT exchange. It is known in the literature as PBE0 [ Adamo99a C. Adamo and V. Barone, “Toward reliable density functional methods without adjustable parameters: The PBE0 model,” J. Chem. Phys., 110 (1999) 6158-69. DOI: ] and as the PBE hybrid [ Ernzerhof99 Ernzerhof, M.; Scuseria, G. E., “Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional,” The Journal of ChemicalPhysics, 1999, 110, 5029-36, DOI: ].
• HSEH1PBE: The recommended version of the full Heyd-Scuseria-Ernzerhof functional, referred to as HSE06 in the literature [ Heyd04 J. Heyd and G. Scuseria, “Efficient hybrid density functional calculations in solids: The HS-Ernzerhof screened Coulomb hybrid functional,” J. Chem. Phys., 121 (2004) 1187-92. DOI: , Heyd04a J. Heyd and G. E. Scuseria, “Assessment and validation of a screened Coulomb hybrid density functional,” J. Chem. Phys., 120 (2004) 7274. DOI: , Heyd05 J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin, “Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional,” J. Chem. Phys., 123 (2005) 174101: 1-8. DOI: , Heyd06 J. Heyd, G. E. Scuseria, and M. Ernzerhof, “Erratum: ‘Hybrid functionals based on a screened Coulomb potential’”, J. Chem. Phys., 124 (2006) 219906. DOI: , Henderson09 T. M. Henderson, A. F. Izmaylov, G. Scalmani, and G. E. Scuseria, “Can short-range hybrids describe long-range-dependent properties?,” J. Chem. Phys., 131 (2009) 044108. DOI: , Izmaylov06 A. F. Izmaylov, G. Scuseria, and M. J. Frisch, “Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems,” J. Chem. Phys., 125 (2006) 104103: 1-8. DOI: , Krukau06 A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, “Influence of the exchange screening parameter on the performance of screened hybrid functionals,” J. Chem. Phys., 125 (2006) 224106. DOI: ].
• OHSE2PBE: The initial form of the HS06 functional, referred to as HSE03 in the literature.
• OHSE1PBE: The version of the HS06 functional prior to modification to support third derivatives.
• PBEh1PBE: Hybrid using the 1998 revised form of PBE pure functional (exchange and correlation) [ Ernzerhof98 M. Ernzerhof and J. P. Perdew, “Generalized gradient approximation to the angle- and system-averaged exchange hole,” J. Chem. Phys., 109 (1998). DOI: ].

Becke One-Parameter Hybrid Functionals

The B1B95 keyword is used to specify Becke’s one-parameter hybrid functional as defined in the original paper [ Becke96 A. D. Becke, “Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing,” J. Chem. Phys., 104 (1996) 1040-46. DOI: ].

The program also provides other, similar one parameter hybrid functionals implemented by Adamo and Barone [ Adamo97 C. Adamo and V. Barone, “Toward reliable adiabatic connection models free from adjustable parameters,” Chem. Phys. Lett., 274 (1997) 242-50. DOI: ]. In one variation, B1LYP, the LYP correlation functional is used (as described for B3LYP above). Another version, mPW1PW91, uses Perdew-Wang exchange as modified by Adamo and Barone combined with PW91 correlation [ Adamo98 C. Adamo and V. Barone, “Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models,” J. Chem. Phys., 108 (1998) 664-75. DOI: ]; the mPW1LYP, mPW1PBE and mPW3PBE variations are available.

Revisions to B97

• Becke’s 1998 revisions to B97 [ Becke97 A. D. Becke, “Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals,” J. Chem. Phys., 107 (1997) 8554-60. DOI: , Schmider98 H. L. Schmider and A. D. Becke, “Optimized density functionals from the extended G2 test set,” J. Chem. Phys., 108 (1998) 9624-31. DOI: ]. The keyword is B98, and it implements fit 2c in reference [ Schmider98 H. L. Schmider and A. D. Becke, “Optimized density functionals from the extended G2 test set,” J. Chem. Phys., 108 (1998) 9624-31. DOI: ].
• Handy, Tozer and coworkers modification to B97: B971 [ Hamprecht98 F. A. Hamprecht, A. Cohen, D. J. Tozer, and N. C. Handy, “Development and assessment of new exchange-correlation functionals,” J. Chem. Phys., 109 (1998) 6264-71. DOI: ].
• Wilson, Bradley and Tozer’s modification to B97: B972 [ Wilson01a P. J. Wilson, T. J. Bradley, and D. J. Tozer, “Hybrid exchange-correlation functional determined from thermochemical data and ab initio potentials,” J. Chem. Phys., 115 (2001) 9233-42. DOI: ].

Functionals with τ-Dependent Gradient-Corrected Correlation

• TPSSh: Hybrid functional using the TPSS functionals [ Tao03 J. M. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, “Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids,” Phys. Rev. Lett., 91 (2003) 146401. DOI: , Staroverov03 V. N. Staroverov, G. E. Scuseria, J. Tao and J. P. Perdew, “Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes,” J. Chem. Phys., 2003, 119, 12129. DOI: ].
• tHCTHhyb: Hybrid functional using the tHCTH functional [ Boese02 A. D. Boese and N. C. Handy, “New exchange-correlation density functionals: The role of the kinetic-energy density,” J. Chem. Phys., 116 (2002) 9559-69. DOI: ].
• BMK: Boese and Martin’s τ-dependent 2004 hybrid functional [ Boese04 A. D. Boese and J. M. L. Martin, “Development of Density Functionals for Thermochemical Kinetics,” J. Chem. Phys., 121 (2004) 3405-16. DOI: ].

Older Functionals

• HISSbPBE requests the HISS functional [ Henderson08 T. M. Henderson, A. F. Izmaylov, G. E. Scuseria and A. Savin, “Assessment of a middle range hybrid functional,” J. Chem. Theory and Comput. 4 (2008) 1254. DOI: ].
• X3LYP: Functional of Xu and Goddard [ Xu04 X. Xu and W. A. Goddard III, “The X3LYP extended density functional for accurate descriptions of nonbond interactions, spin states, and thermochemical properties,” Proc. Natl. Acad. Sci. USA, 101 (2004) 2673-77. DOI: ].

Half-and-Half Functionals

The following functionals, which are included for backward-compatibility only. Note that these are not the same as the “half-and-half” functionals proposed by Becke [ Becke93 A. D. Becke, “A new mixing of Hartree-Fock and local density-functional theories,” J. Chem. Phys., 98 (1993) 1372-77. DOI: ].

• BHandH: 0.5*E_X^HF + 0.5*E_X^LSDA + E_C^LYP
• BHandHLYP: 0.5*E_X^HF + 0.5*E_X^LSDA + 0.5*ΔE_X^Becke88 + E_C^LYP

User-Defined Hybrid Models

Gaussian 16 can use any model of the general form:

P₂E_X^HF + P₁(P₄E_X^Slater + P₃ΔE_x^non-local) + P₆E_C^local + P₅ΔE_C^non-local

The only available local exchange method is Slater (S), which should be used when only local exchange is desired. Any combinable non-local exchange functional and combinable correlation functional may be used (as listed previously).

The values of the six parameters are specified with various non-standard options to the program:

• IOp(3/76=mmmmmnnnnn) sets P₁ to mmmmm/10000 and P₂ to nnnnn/10000. P₁ is usually set to either 1.0 or 0.0, depending on whether an exchange functional is desired or not, and any scaling is accomplished using P₃ and P₄.
• IOp(3/77=mmmmmnnnnn) sets P₃ to mmmmm/10000 and P₄ to nnnnn/10000.
• IOp(3/78=mmmmmnnnnn) sets P₅ to mmmmm/10000 and P₆ to nnnnn/10000.

For example, IOp(3/76=1000005000) sets P₁ to 1.0 and P₂ to 0.5. Note that all values must be expressed using five digits, adding any necessary leading zeros.

Here is a route section specifying the functional corresponding to the B3LYP keyword:

#P BLYP IOp(3/76=1000002000) IOp(3/77=0720008000) IOp(3/78=0810010000)

The output file displays the values that are in use:

 IExCor=  402 DFT=T Ex=B+HF Corr=LYP ExCW=0 ScaHFX=  0.200000
                ScaDFX=  0.800000  0.720000  1.000000  0.810000 
                

where the value of ScaHFX is P₂, and the sequence of values given for ScaDFX are P₄, P₃, P₆, and P₅.

Pure関数キーワード

Keywords: Pure Functionals

Names for the various pure DFT models are given by combining the names for the exchange and correlation functionals. In some cases, standard synonyms used in the field are also available as keywords. In order to specify a pure functional, combine an exchange functional component keyword with the one for desired correlation functional. For example, the combination of the Becke exchange functional (B) and the LYP correlation functional is requested by the BLYP keyword. Similarly, SVWN requests the Slater exchange functional (S) and the VWN correlation functional, and is known in the literature by its synonym LSDA (Local Spin Density Approximation). LSDA is a synonym for SVWN. Some other software packages with DFT facilities use the equivalent of SVWN5 when “LSDA” is requested. Check the documentation carefully for all packages when making comparisons.

Exchange Functionals

The following exchange functionals are available in Gaussian 16. Unless otherwise indicated, these exchange functionals must be combined with a correlation functional in order to produce a usable method.

• S: The Slater exchange, ρ^4/3 with theoretical coefficient of 2/3, also referred to as Local Spin Density exchange [ Hohenberg64 P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Phys. Rev., 136 (1964) B864-B71. DOI: , Kohn65 W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev., 140 (1965) A1133-A38. DOI: , Slater74]. Keyword if used alone: HFS.
• XA: The XAlpha exchange, ρ^4/3 with the empirical coefficient of 0.7, usually employed as a standalone exchange functional, without a correlation functional [ Hohenberg64 P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Phys. Rev., 136 (1964) B864-B71. DOI: , Kohn65 W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev., 140 (1965) A1133-A38. DOI: , Slater74]. Keyword if used alone: XAlpha.
• B: Becke’s 1988 functional, which includes the Slater exchange along with corrections involving the gradient of the density [ Becke88b A. D. Becke, “Density-functional exchange-energy approximation with correct asymptotic-behavior,” Phys. Rev. A, 38 (1988) 3098-100. DOI: ]. Keyword if used alone: HFB.
• PW91: The exchange component of Perdew and Wang’s 1991 functional [Perdew91, Perdew92 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, “Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, 46 (1992) 6671-87. DOI: , Perdew93a J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, “Erratum: Atoms, molecules, solids, and surfaces - Applications of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, 48 (1993) 4978. DOI: , Perdew96 J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-correlation hole of a many-electron system,” Phys. Rev. B, 54 (1996) 16533-39. DOI: , Burke98].
• mPW: The Perdew-Wang 1991 exchange functional as modified by Adamo and Barone [ Adamo98 C. Adamo and V. Barone, “Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models,” J. Chem. Phys., 108 (1998) 664-75. DOI: ].
• G96: The 1996 exchange functional of Gill [ Gill96 P. M. W. Gill, “A new gradient-corrected exchange functional,” Mol. Phys., 89 (1996) 433-45. DOI: , Adamo98a C. Adamo and V. Barone, “Implementation and validation of the Lacks-Gordon exchange functional in conventional density functional and adiabatic connection methods,” J. Comp. Chem., 19 (1998) 418-29. DOI: ].
• PBE: The 1996 functional of Perdew, Burke and Ernzerhof [ Perdew96a J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett., 77 (1996) 3865-68. DOI: , Perdew97 J. P. Perdew, K. Burke, and M. Ernzerhof, “Errata: Generalized gradient approximation made simple,” Phys. Rev. Lett., 78 (1997) 1396. DOI: ].
• O: Handy’s OPTX modification of Becke’s exchange functional [ Handy01 N. C. Handy and A. J. Cohen, “Left-right correlation energy,” Mol. Phys., 99 (2001) 403-12. DOI: , Hoe01 W.-M. Hoe, A. Cohen, and N. C. Handy, “Assessment of a new local exchange functional OPTX,” Chem. Phys. Lett., 341 (2001) 319-28. DOI: ].
• TPSS: The exchange functional of Tao, Perdew, Staroverov, and Scuseria [ Tao03 J. M. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, “Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids,” Phys. Rev. Lett., 91 (2003) 146401. DOI: ].
• RevTPSS: The revised TPSS exchange functional of Perdew et. al. [ Perdew09 John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Lucian A. Constantin, and Jianwei Sun, “Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry,” Phys. Rev. Lett. 103 (2009) 026403. DOI: , Perdew11 John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Lucian A. Constantin, and Jianwei Sun, “Erratum: ‘Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry’ [Phys. Rev. Lett. 103, 026403 (2009)]” Phys. Rev. Lett. 106 (2011) 179902(E). DOI: ].
• BRx: The 1989 exchange functional of Becke [ Becke89a A. D. Becke and M. R. Roussel, “Exchange holes in inhomogeneous systems: A coordinate-space model,” Phys. Rev. A, 39 (1989) 3761-67. DOI: ].
• PKZB: The exchange part of the Perdew, Kurth, Zupan and Blaha functional [ Perdew99 J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, “Accurate density functional with correct formal properties: A step beyond the generalized gradient approximation,” Phys. Rev. Lett., 82 (1999) 2544-47. DOI: ].
• wPBEh: The exchange part of screened Coulomb potential-based final of Heyd, Scuseria and Ernzerhof (also known as HSE) [ Heyd03 J. Heyd, G. Scuseria, and M. Ernzerhof, “Hybrid functionals based on a screened Coulomb potential,” J. Chem. Phys., 118 (2003) 8207-15. DOI: , Izmaylov06 A. F. Izmaylov, G. Scuseria, and M. J. Frisch, “Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems,” J. Chem. Phys., 125 (2006) 104103: 1-8. DOI: , Henderson09 T. M. Henderson, A. F. Izmaylov, G. Scalmani, and G. E. Scuseria, “Can short-range hybrids describe long-range-dependent properties?,” J. Chem. Phys., 131 (2009) 044108. DOI: ].
• PBEh: 1998 revision of PBE [ Ernzerhof98 M. Ernzerhof and J. P. Perdew, “Generalized gradient approximation to the angle- and system-averaged exchange hole,” J. Chem. Phys., 109 (1998). DOI: ].

Correlation Functionals

The following correlation functionals are available, listed by their corresponding keyword component, all of which must be combined with the keyword for the desired exchange functional:

• VWN: Vosko, Wilk, and Nusair 1980 correlation functional(III) fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation [ Vosko80 S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis,” Can. J. Phys., 58 (1980) 1200-11. DOI: ] (functional III in this article).
• VWN5: Functional V from reference [ Vosko80 S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis,” Can. J. Phys., 58 (1980) 1200-11. DOI: ] which fits the Ceperly-Alder solution to the uniform electron gas (this is the functional recommended in [ Vosko80 S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis,” Can. J. Phys., 58 (1980) 1200-11. DOI: ]).
• LYP: The correlation functional of Lee, Yang, and Parr, which includes both local and non-local terms [ Lee88 C. Lee, W. Yang, and R. G. Parr, “Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density,” Phys. Rev. B, 37 (1988) 785-89. DOI: , Miehlich89 B. Miehlich, A. Savin, H. Stoll, and H. Preuss, “Results obtained with the correlation-energy density functionals of Becke and Lee, Yang and Parr,” Chem. Phys. Lett., 157 (1989) 200-06. DOI: ].
• PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981) [ Perdew81 J. P. Perdew and A. Zunger, “Self-interaction correction to density-functional approximations for many-electron systems,” Phys. Rev. B, 23 (1981) 5048-79. DOI: ].
• P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local correlation functional [ Perdew86 J. P. Perdew, “Density-functional approximation for the correlation energy of the inhomogeneous electron gas,” Phys. Rev. B, 33 (1986) 8822-24. DOI: ].
• PW91 (Perdew/Wang 91): Perdew and Wang’s 1991 gradient-corrected correlation functional [Perdew91, Perdew92 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, “Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, 46 (1992) 6671-87. DOI: , Perdew93a J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, “Erratum: Atoms, molecules, solids, and surfaces - Applications of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, 48 (1993) 4978. DOI: , Perdew96 J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-correlation hole of a many-electron system,” Phys. Rev. B, 54 (1996) 16533-39. DOI: , Burke98].
• B95 (Becke 95): Becke’s τ-dependent gradient-corrected correlation functional (defined as part of his one parameter hybrid functional [ Becke96 A. D. Becke, “Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing,” J. Chem. Phys., 104 (1996) 1040-46. DOI: ]).
• PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and Ernzerhof [ Perdew96a J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett., 77 (1996) 3865-68. DOI: , Perdew97 J. P. Perdew, K. Burke, and M. Ernzerhof, “Errata: Generalized gradient approximation made simple,” Phys. Rev. Lett., 78 (1997) 1396. DOI: ].
• TPSS: The τ-dependent gradient-corrected functional of Tao, Perdew, Staroverov, and Scuseria [ Tao03 J. M. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, “Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids,” Phys. Rev. Lett., 91 (2003) 146401. DOI: ].
• RevTPSS: The revised TPSS correlation functional of Perdew et. al. [ Perdew09 John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Lucian A. Constantin, and Jianwei Sun, “Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry,” Phys. Rev. Lett. 103 (2009) 026403. DOI: , Perdew11 John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Lucian A. Constantin, and Jianwei Sun, “Erratum: ‘Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry’ [Phys. Rev. Lett. 103, 026403 (2009)]” Phys. Rev. Lett. 106 (2011) 179902(E). DOI: ].
• KCIS: The Krieger-Chen-Iafrate-Savin correlation functional [ Rey98 J. Rey and A. Savin, “Virtual space level shifting and correlation energies,” Int. J. Quantum Chem., 69 (1998) 581-90. DOI: , Krieger99 J. B. Krieger, J. Q. Chen, G. J. Iafrate, and A. Savin, in Electron Correlations and Materials Properties, Ed. A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, New York, 1999) 463-77. DOI: , Krieger01 J. B. Krieger, J. Q. Chen, and S. Kurth, in Density Functional Theory and its Application to Materials, Ed. V. VanDoren, C. VanAlsenoy, and P. Geerlings, A.I.P. Conference Proceedings, Vol. 577 (A.I.P., New York, 2001) 48-69. DOI: , Toulouse02 J. Toulouse, A. Savin, and C. Adamo, “Validation and assessment of an accurate approach to the correlation problem in density functional theory: The Krieger-Chen-Iafrate-Savin model,” J. Chem. Phys., 117 (2002) 10465-73. DOI: ].
• BRC: Becke-Roussel correlation functional [ Becke89a A. D. Becke and M. R. Roussel, “Exchange holes in inhomogeneous systems: A coordinate-space model,” Phys. Rev. A, 39 (1989) 3761-67. DOI: ].
• PKZB: The correlation part of the Perdew, Kurth, Zupan and Blaha functional [ Perdew99 J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, “Accurate density functional with correct formal properties: A step beyond the generalized gradient approximation,” Phys. Rev. Lett., 82 (1999) 2544-47. DOI: ].

Correlation Functional Variations. The following correlation functionals combine local and non-local terms from different correlation functionals:

• VP86: VWN5 local and P86 non-local correlation functional.
• V5LYP: VWN5 local and LYP non-local correlation functional.

Standalone Pure Functionals

The following pure functionals are self-contained and are not combined with any other functional keyword components:

• VSXC: van Voorhis and Scuseria’s τ-dependent gradient-corrected correlation functional [ VanVoorhis98 T. Van Voorhis and G. E. Scuseria, “A novel form for the exchange-correlation energy functional,” J. Chem. Phys., 109 (1998) 400-10. DOI: ].
• HCTH/*: Handy’s family of functionals including gradient-corrected correlation [ Hamprecht98 F. A. Hamprecht, A. Cohen, D. J. Tozer, and N. C. Handy, “Development and assessment of new exchange-correlation functionals,” J. Chem. Phys., 109 (1998) 6264-71. DOI: , Boese00 A. D. Boese, N. L. Doltsinis, N. C. Handy, and M. Sprik, “New generalized gradient approximation functionals,” J. Chem. Phys., 112 (2000) 1670-78. DOI: , Boese01 A. D. Boese and N. C. Handy, “A new parametrization of exchange-correlation generalized gradient approximation functionals,” J. Chem. Phys., 114 (2001) 5497-503. DOI: ]. HCTH refers to HCTH/407, HCTH93 to HCTH/93, HCTH147 to HCTH/147, and HCTH407 to HCTH/407. Note that the related HCTH/120 functional is not implemented.
• tHCTH: The τ-dependent member of the HCTH family [ Boese02 A. D. Boese and N. C. Handy, “New exchange-correlation density functionals: The role of the kinetic-energy density,” J. Chem. Phys., 116 (2002) 9559-69. DOI: ]. See also tHCTHhyb.
• B97D: Grimme’s functional including dispersion [ Grimme06 S. Grimme, “Semiempirical GGA-type density functional constructed with a long-range dispersion correction,” J. Comp. Chem., 27 (2006) 1787-99. DOI: ]. B97D3 requests the same but with Grimme’s D3BJ dispersion [ Grimme11 S. Grimme, S. Ehrlich and L. Goerigk, “Effect of the damping function in dispersion corrected density functional theory,” J. Comp. Chem. 32 (2011) 1456-65. DOI: ].
• M06L [ Zhao06a Y. Zhao and D. G. Truhlar, “A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions,” J. Chem. Phys., 125 (2006), 194101: 1-18. DOI: ], SOGGA11 [ Peverati11 R. Peverati, Y. Zhao and D. G. Truhlar, “Generalized Gradient Approximation That Recovers the Second-Order Density-Gradient Expansion with Optimized Across-the-Board Performance,” J. Phys. Chem. Lett. 2 (2011) 1991-1997. DOI: ], M11L [ Peverati12 R. Peverati and D. G. Truhlar, “M11-L: A Local Density Functional That Provides Improved Accuracy for Electronic Structure Calculations in Chemistry and Physics,” J. Phys. Chem. Lett. 3 (2012) 117-124. DOI: ], MN12L [ Peverati12c R. Peverati and D. G. Truhlar, “An improved and broadly accurate local approximation to the exchange–correlation density functional: The MN12-L functional for electronic structure calculations in chemistry and physics,” Phys. Chem. Chem. Phys. 10 (2012) 13171. DOI: ] N12 [ Peverati12b R. Peverati and D. G. Truhlar, “Exchange-Correlation Functional with Good Accuracy for Both Structural and Energetic Properties while Depending Only on the Density and Its Gradient,” J. Chem. Theory and Comput. 8 (2012) 2310-2319. DOI: ] and MN15L [ Yu16a H. S. Yu, X. He, and D. G. Truhlar, “MN15-L: A New Local Exchange-Correlation Functional for Kohn–Sham Density Functional Theory with Broad Accuracy for Atoms, Molecules, and Solids,” Journal of Chemical Theory and Computation 2016, 12, 1280-1293. DOI: ] request these pure functionals from the Truhlar group.

経験的分散力

Dispersion

The EmpiricalDispersion keyword enables empirical dispersion. It takes the following options:

PFD

Add the Petersson-Frisch dispersion model from the APFD functional [ Austin12 A. Austin, G. Petersson, M. J. Frisch, F. J. Dobek, G. Scalmani, and K. Throssell, “A density functional with spherical atom dispersion terms,” J. Chem. Theory and Comput. 8 (2012) 4989. DOI: ].

GD2

Add the D2 version of Grimme’s dispersion [ Grimme06 S. Grimme, “Semiempirical GGA-type density functional constructed with a long-range dispersion correction,” J. Comp. Chem., 27 (2006) 1787-99. DOI: ]. The table below gives the list of functionals in Gaussian 16 for which GD2 parameters are defined. The functionals highlighted in bold include this dispersion model by default when the indicated keyword is specified (e.g., B2PLYPD). For the rest of the functionals, dispersion is requested with EmpiricalDispersion=GD2.

The damping function used by this model also contains a D6 parameter with a fixed value of 6.0.

You can use this empirical dispersion method with other functionals via the IOps(3/174,176) (SR6 should be 1.1).

The wB97XD functional specified as an independent keyword uses a version of this dispersion model with values of S6 and SR6 of 1.0 and 1.1, respectively. This functional uses a similar damping function to that used by the GD3 model, with D6 and IA6 having fixed values of 6.0 and 12, respectively.

GD3

Add the D3 version of Grimme’s dispersion with the original D3 damping function [ Grimme10 S. Grimme, J. Antony, S. Ehrlich and H. Krieg, “A consistent and accurate ab initio parameterization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu,” J. Chem. Phys., 132 (2010) 154104. DOI: ]. The table below gives the list of functionals in Gaussian 16 for which GD3 parameters are defined. For these functionals, dispersion is requested with EmpiricalDispersion=GD3.

This model also uses an SR8 parameter with a fixed value of 1.0. The damping function used by this model also contains D6, IA6, D8, and IA8 parameters with fixed values of 6.0, 14, 6.0, and 16, respectively.

You can use this empirical dispersion method with other functionals via the IOps(3/174-176) (S6 should be 1.0).

GD3BJ

Add the D3 version of Grimme’s dispersion with Becke-Johnson damping [ Grimme11 S. Grimme, S. Ehrlich and L. Goerigk, “Effect of the damping function in dispersion corrected density functional theory,” J. Comp. Chem. 32 (2011) 1456-65. DOI: ]. The table below gives the list of functionals in Gaussian 16 for which GD3BJ parameters are defined. The functionals highlighted in bold include this dispersion model by default when the indicated keyword is specified (e.g., B2PLYPD3). For the rest of the functionals, dispersion is requested with EmpiricalDispersion=GD3BJ.

You can use this empirical dispersion method with other functionals via the IOps(3/174-178) (S6 should be 1.0).

適用範囲

Availability

Energies, analytic gradients, and analytic frequencies; ADMP calculations.

Third order properties such as hyperpolarizabilities and Raman intensities are not available for functionals for which third derivatives are not implemented: the exchange functionals G96, P86, PKZB, wPBEh and PBEh; the correlation functional PKZB; the hybrid functionals OHSE1PBE and OHSE2PBE.

関連キーワード

Related Keywords

IOp, Int=Grid , Stable , TD , DenFit , B2PLYP , mPW2LYP

実例

Examples

The energy is reported in DFT calculations in a form similar to that of Hartree-Fock calculations. Here is the energy output from a B3LYP calculation:

 SCF Done:  E(RB3LYP) =  -75.3197099428     A.U. after    5 cycles