Adiabatic Connection Forms in DFT

David J Tozerwww.dur.ac.uk/d.j.tozer

1 / 17

The Adiabatic Connection

Hλ = −1

2

Xi

∇2i + λVee +

Xi

vλ(ri )

λ = 0, Non-interacting λ = 1, Interacting system

EXC =

Z 1

0

Wλdλ

Wλ = 〈ψλ|Vee|ψλ〉 − J

0

0

W0

EXC

−TC

λ

W1

1

Wλ

2 / 17

The Adiabatic Connection

Hλ = −1

2

Xi

∇2i + λVee +

Xi

vλ(ri )

λ = 0, Non-interacting λ = 1, Interacting system

EXC =

Z 1

0

Wλdλ

Wλ = 〈ψλ|Vee|ψλ〉 − J

0

0

W0

EXC

−TC

λ

W1

1

Wλ

2 / 17

Exact Properties

0

0

W0

EXC

−TC

λ

W1

1

Wλ Wλ = 〈ψλ|Vee|ψλ〉 − J

W0 = 〈ψ0|Vee|ψ0〉 − J = J + EX − J = EX

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

∂λ

˛λ=0

= 2EC,GL2 = −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj− 2

Xi,α

〈φi |vX − vNLX |φα〉2

εα − εi

3 / 17

Exact Properties

0

0

W0

EXC

−TC

λ

W1

1

Wλ Wλ = 〈ψλ|Vee|ψλ〉 − J

W0 = 〈ψ0|Vee|ψ0〉 − J = J + EX − J = EX

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

∂λ

˛λ=0

= 2EC,GL2 = −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj− 2

Xi,α

〈φi |vX − vNLX |φα〉2

εα − εi

3 / 17

Exact Properties

0

0

W0

EXC

−TC

λ

W1

1

Wλ Wλ = 〈ψλ|Vee|ψλ〉 − J

W0 = 〈ψ0|Vee|ψ0〉 − J = J + EX − J = EX

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

∂λ

˛λ=0

= 2EC,GL2 = −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj− 2

Xi,α

〈φi |vX − vNLX |φα〉2

εα − εi

3 / 17

Exact Properties

0

0

W0

EXC

−TC

λ

W1

1

Wλ Wλ = 〈ψλ|Vee|ψλ〉 − J

W0 = 〈ψ0|Vee|ψ0〉 − J = J + EX − J = EX

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

∂λ

˛λ=0

= 2EC,GL2 = −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj− 2

Xi,α

〈φi |vX − vNLX |φα〉2

εα − εi

3 / 17

Functional Development ...

A new mixing of Hartree-Fock and local density-functional theories

Axe1 D. Becke Department of Chemistry, Queen’s University, Kingston, Ontario, Canada K7L 3N6

(Received 12 August 1992; accepted 8 October 1992)

Previous attempts to combine Hartree-Fock theory with local density-functional theory have been unsuccessful in applications to molecular bonding. We derive a new coupling of these two theories that maintains their simplicity and computational efficiency, and yet greatly improves their predictive power. Very encouraging results of tests on atomization energies, ionization potentials, and proton affinities are reported, and the potential for future development is discussed.

I. INTRODUCTION

There has long been interest in the problem of coupling Hartree-Fock theory with local density-functional approx- imations for dynamical correlation (see Ref. 1 and refer- ences therein). In doing so, one attempts to exploit the obvious strengths of each partner. Hartree-Fock theory provides an exact treatment of exchange at a cost that scales well with molecular size and is a practical compu- tational tool even for large chemical systems. Unfortu- nately, it suffers well-known and severe deficiencies in de- scribing chemical bonding and cannot be used in thermochemical applications, excepting isodesmic reac- tions, without further corrections for “correlation.” These post-Hartree-Fock corrections (i.e., Moller-Plesset per- turbation theory, configuration interaction, etc.) do not scale well with molecular size and are, at least currently, impractical in large systems.

On the other hand, local density-functional correlation approximations can be evaluated extremely easily and quickly (i.e., by numerical integration of functionals de- pending only on total electronic density) and appear to offer a convenient alternative to conventional post- Hartree+Fock technology. Straightforward addition of electron-gas correlation approximations to Hartree-Fock energies has not, however, proven successful in thermo- chemical tests,’ and more sophisticated approaches have consequently been proposed. Unfortunately, these involve multiconfiguration self-consistent field (MCSCF) (Refs. 2-5) or generalized valence-bond (GVB) (Ref. 6) refer- ence states and, hence, the beauty and practicality of the original “Hartree-Fock plus density-functionals” idea is lost.

In the present work, we propose a new mixing of HartreeFock theory and local density-functional theory that performs surprisingly well on the thermochemical tests of Pople’s Gaussian-l (Gl ) data base.7-9 This new approach is no more costly than its predecessors, but de- livers superior accuracy and holds great promise for fur- ther development.

II. THEORETICAL BACKGROUND

The analysis of the present work will be phrased in the language of the Kohn-Sham formulation of density- functional theory. lo Other formulations are also possible

but will not be considered here.” We should note at the outset that the term ‘LHartree-Fock” theory is not formally appropriate in the Kohn-Sham density-functional context, and that the term “exact exchange” is preferable. The dis- tinctions are subtle, however, and will be dealt with at appropriate points in the subsequent text.

In Kohn-Sham density-functional theory, we imagine a reference system of ultimate simplicity: a system of inde- pendent noninteracting electrons in a common, one-body potential Vx, yielding the same density as the “real” fully- interacting system. To be more specific, we imagine a set of independent reference orbitals pi satisfying the following independent-particle Schrbdinger equation:

-!V2+j+ VKS*i=ef*i (1)

with a local one-body potential V,, defined such that the noninteracting density

p= 5 l$il” (2) i

equals the density of the “real” system. Then, we express the total electronic energy of the real, fully-interacting sys- tem as

E total= T, + s

P Vnucd3r

pWp(r2) d3rld3r2 + Exe , (3)

r12

where To is the kinetic energy of the noninteracting refer- ence system, the second and third terms are the nuclear interaction energy and the classical Coulomb self-energy, respectively, and the last term E,, is the density-functional “exchange-correlation” energy.

Equation (3), in fact, defines the Kohn-Sham exchange-correlation energy. Although innocuous in ap- pearance, Exe contains, buried within it, all the details of two-body exchange and dynamical correlation and a kinetic-energy component as well. Nevertheless, it can be shown” that E,, depends uniquely on the total electronic density distribution and that the Kohn-Sham local poten- tial in Eq. ( 1) is given by

vKS = vn”c + vel + vXC 2 (4)

where Vel is the usual electronic Coulomb potential,

1372 J. Chem. Phys. 98 (2), 15 January 1993 0021-9606/93/021372-6$06.00 @ 1993 American Institute of Physics

Downloaded 27 Sep 2002 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

Wλ = a + bλ EXC = a +b

2H&H→ B3LYP

Self-interaction-free exchange-correlation functional for thermochemistryand kinetics

Paula Mori-Sánchez,a! Aron J. Cohen,a! and Weitao YangDepartment of Chemistry, Duke University, Durham, North Carolina 27708

!Received 20 January 2006; accepted 30 January 2006; published online 3 March 2006"

We develop a self-interaction-free exchange-correlation functional which is very accurate forthermochemistry and kinetics. This is achieved by theoretical construction of the functional formand nonlinear fitting. We define a simple interpolation of the adiabatic connection that uses exactexchange, generalized gradient approximation !GGA" and meta-GGA functionals. The performanceis optimized by fitting a small number of empirical parameters. Overall the new functional improvessignificantly upon hybrids and meta-GGAs while correctly describing one-electron systems. Themean absolute error on a large set of reaction barriers is reduced to 1.99 kcal/mol. © 2006American Institute of Physics. #DOI: 10.1063/1.2179072$

The theoretical formulation and implementation of im-proved exchange-correlation energy functionals is at theheart of density-functional theory !DFT" development.1 Im-pressive progress has been made from the early local densityapproximation !LDA"2 to generalized gradient approxima-tions !GGAs"3–5 and from GGAs to more recent implicitfunctionals constructed from the KS orbitals. Hybrid func-tionals which include a fraction of exact exchange6 andmeta-GGAs which make use of orbital kinetic energydensities,7 have led to the high accuracy and the wide usageof DFT. Despite their success, these functionals still exhibitimportant failures. For instance, B3LYP,5,6 the most popularof present functionals, gives errors greater than 4.0 kcal/molfor reaction energy barriers. Higher accuracy is needed.

Many of the remaining problems of current DFT ap-proximations are associated with the self-interaction error!SIE". An electron does not interact with itself. In a one-electron system, this leads to the requirement that the ex-change energy exactly cancels the Coulomb energy and thecorrelation energy vanishes,

Ex#!$ = ! J#!$; Ec#!$ = 0, !1"

where ! denotes any one-electron density. Availableexchange-correlation functionals violate one or both condi-tions, perhaps with the exception of Becke’s recent work.8

The self-interaction error is undoubtedly the origin ofmany qualitative and quantitative failures which plague DFT.Some of these are dissociation of one-electron systems andradicals, treatment of fractional number of electrons, chargetransfer processes, underestimation of reaction barriers, andover estimation of polarizabilities.9,10

There have been two main avenues for SIE correctionwhich have not been totally successful. One case is exempli-fied by hybrid functionals, which only partially reduce theSIE. Another approach is based on the straightforward re-moval of the SIE by using Perdew-Zunger correction.11

However, such SIE corrected functionals are inferior to their

parent functionals in terms of performance.12 For these rea-sons the correction of the SIE, while maintaining high accu-racy, remains a very important challenge in DFT.

In this Communication, we address the SIE problemwithin Kohn-Sham !KS" DFT.13 The adiabatic connection!AC" is modeled using exact exchange, GGAs and meta-GGAs, which gives a new exchange-correlation energy func-tional that is exact for one-electron densities. The functionalis accurate for thermochemistry and kinetics, while havingmany attractive formal properties. We will show that overallthe new functional, at similar computational cost, signifi-cantly improves upon commonly used functionals such asBLYP or B3LYP.

Our theoretical framework is the adiabatic connection,which has guided and illuminated the construction ofexchange-correlation functionals for many years.14–18 Con-sider a family of N-electron systems, Hv!""= T+ Vee!rN ,""+%i,#

N v#!ri ,"", where the electron-electron interaction andthe external potential are parametrized in terms of ", "! #0,1$. We focus on the linear adiabatic connection,Vee!rN ,""=%i$j" /rij, that links the KS system of noninter-acting particles !"=0" with the physical system !"=1". Theexchange-correlation energy is exactly defined by thecoupling-constant integration formula

Exc#!$ = &0

1

W"#!$d" , !2"

where W"#!$= '%"(Vee(%")!J#!$. Here J#!$ is the Coulombenergy, J#!$=1/2*!!r"!!r!" / (r!r!(drdr!, and %" is theunique antisymmetric ground state wave function of H", as-suming v-representability. Following Langreth and Perdew14

the electron density is fixed along the path, !!r"=!0!r"=!"!r""", such that %"!r" yields !!r" and minimizes theexpectation value 'T+"Vee). Once the integrand of the adia-batic connection is known, Exc is obtained from Eq. !2".

The challenge of the AC approach is that the exact W" isunknown. Thus, our main goal is to build accurate approxi-mations of W". In contrast to the usual density functionala"Both authors contributed equally to this work.

THE JOURNAL OF CHEMICAL PHYSICS 124, 091102 !2006"

0021-9606/2006/124"9!/091102/4/$23.00 © 2006 American Institute of Physics124, 091102-1

Downloaded 06 Mar 2006 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Wλ = a +bλ

1 + cλEXC = a +

b

c

“1− loge(1 + c)

c

”MCY

(Ernzerhof, CPL 263 499 (1996))

4 / 17

Functional Development ...

A new mixing of Hartree-Fock and local density-functional theories

Axe1 D. Becke Department of Chemistry, Queen’s University, Kingston, Ontario, Canada K7L 3N6

(Received 12 August 1992; accepted 8 October 1992)

Previous attempts to combine Hartree-Fock theory with local density-functional theory have been unsuccessful in applications to molecular bonding. We derive a new coupling of these two theories that maintains their simplicity and computational efficiency, and yet greatly improves their predictive power. Very encouraging results of tests on atomization energies, ionization potentials, and proton affinities are reported, and the potential for future development is discussed.

I. INTRODUCTION

There has long been interest in the problem of coupling Hartree-Fock theory with local density-functional approx- imations for dynamical correlation (see Ref. 1 and refer- ences therein). In doing so, one attempts to exploit the obvious strengths of each partner. Hartree-Fock theory provides an exact treatment of exchange at a cost that scales well with molecular size and is a practical compu- tational tool even for large chemical systems. Unfortu- nately, it suffers well-known and severe deficiencies in de- scribing chemical bonding and cannot be used in thermochemical applications, excepting isodesmic reac- tions, without further corrections for “correlation.” These post-Hartree-Fock corrections (i.e., Moller-Plesset per- turbation theory, configuration interaction, etc.) do not scale well with molecular size and are, at least currently, impractical in large systems.

On the other hand, local density-functional correlation approximations can be evaluated extremely easily and quickly (i.e., by numerical integration of functionals de- pending only on total electronic density) and appear to offer a convenient alternative to conventional post- Hartree+Fock technology. Straightforward addition of electron-gas correlation approximations to Hartree-Fock energies has not, however, proven successful in thermo- chemical tests,’ and more sophisticated approaches have consequently been proposed. Unfortunately, these involve multiconfiguration self-consistent field (MCSCF) (Refs. 2-5) or generalized valence-bond (GVB) (Ref. 6) refer- ence states and, hence, the beauty and practicality of the original “Hartree-Fock plus density-functionals” idea is lost.

In the present work, we propose a new mixing of HartreeFock theory and local density-functional theory that performs surprisingly well on the thermochemical tests of Pople’s Gaussian-l (Gl ) data base.7-9 This new approach is no more costly than its predecessors, but de- livers superior accuracy and holds great promise for fur- ther development.

II. THEORETICAL BACKGROUND

The analysis of the present work will be phrased in the language of the Kohn-Sham formulation of density- functional theory. lo Other formulations are also possible

but will not be considered here.” We should note at the outset that the term ‘LHartree-Fock” theory is not formally appropriate in the Kohn-Sham density-functional context, and that the term “exact exchange” is preferable. The dis- tinctions are subtle, however, and will be dealt with at appropriate points in the subsequent text.

In Kohn-Sham density-functional theory, we imagine a reference system of ultimate simplicity: a system of inde- pendent noninteracting electrons in a common, one-body potential Vx, yielding the same density as the “real” fully- interacting system. To be more specific, we imagine a set of independent reference orbitals pi satisfying the following independent-particle Schrbdinger equation:

-!V2+j+ VKS*i=ef*i (1)

with a local one-body potential V,, defined such that the noninteracting density

p= 5 l$il” (2) i

equals the density of the “real” system. Then, we express the total electronic energy of the real, fully-interacting sys- tem as

E total= T, + s

P Vnucd3r

pWp(r2) d3rld3r2 + Exe , (3)

r12

where To is the kinetic energy of the noninteracting refer- ence system, the second and third terms are the nuclear interaction energy and the classical Coulomb self-energy, respectively, and the last term E,, is the density-functional “exchange-correlation” energy.

Equation (3), in fact, defines the Kohn-Sham exchange-correlation energy. Although innocuous in ap- pearance, Exe contains, buried within it, all the details of two-body exchange and dynamical correlation and a kinetic-energy component as well. Nevertheless, it can be shown” that E,, depends uniquely on the total electronic density distribution and that the Kohn-Sham local poten- tial in Eq. ( 1) is given by

vKS = vn”c + vel + vXC 2 (4)

where Vel is the usual electronic Coulomb potential,

1372 J. Chem. Phys. 98 (2), 15 January 1993 0021-9606/93/021372-6$06.00 @ 1993 American Institute of Physics

Downloaded 27 Sep 2002 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

Wλ = a + bλ EXC = a +b

2H&H→ B3LYP

Self-interaction-free exchange-correlation functional for thermochemistryand kinetics

Paula Mori-Sánchez,a! Aron J. Cohen,a! and Weitao YangDepartment of Chemistry, Duke University, Durham, North Carolina 27708

!Received 20 January 2006; accepted 30 January 2006; published online 3 March 2006"

We develop a self-interaction-free exchange-correlation functional which is very accurate forthermochemistry and kinetics. This is achieved by theoretical construction of the functional formand nonlinear fitting. We define a simple interpolation of the adiabatic connection that uses exactexchange, generalized gradient approximation !GGA" and meta-GGA functionals. The performanceis optimized by fitting a small number of empirical parameters. Overall the new functional improvessignificantly upon hybrids and meta-GGAs while correctly describing one-electron systems. Themean absolute error on a large set of reaction barriers is reduced to 1.99 kcal/mol. © 2006American Institute of Physics. #DOI: 10.1063/1.2179072$

The theoretical formulation and implementation of im-proved exchange-correlation energy functionals is at theheart of density-functional theory !DFT" development.1 Im-pressive progress has been made from the early local densityapproximation !LDA"2 to generalized gradient approxima-tions !GGAs"3–5 and from GGAs to more recent implicitfunctionals constructed from the KS orbitals. Hybrid func-tionals which include a fraction of exact exchange6 andmeta-GGAs which make use of orbital kinetic energydensities,7 have led to the high accuracy and the wide usageof DFT. Despite their success, these functionals still exhibitimportant failures. For instance, B3LYP,5,6 the most popularof present functionals, gives errors greater than 4.0 kcal/molfor reaction energy barriers. Higher accuracy is needed.

Many of the remaining problems of current DFT ap-proximations are associated with the self-interaction error!SIE". An electron does not interact with itself. In a one-electron system, this leads to the requirement that the ex-change energy exactly cancels the Coulomb energy and thecorrelation energy vanishes,

Ex#!$ = ! J#!$; Ec#!$ = 0, !1"

where ! denotes any one-electron density. Availableexchange-correlation functionals violate one or both condi-tions, perhaps with the exception of Becke’s recent work.8

The self-interaction error is undoubtedly the origin ofmany qualitative and quantitative failures which plague DFT.Some of these are dissociation of one-electron systems andradicals, treatment of fractional number of electrons, chargetransfer processes, underestimation of reaction barriers, andover estimation of polarizabilities.9,10

There have been two main avenues for SIE correctionwhich have not been totally successful. One case is exempli-fied by hybrid functionals, which only partially reduce theSIE. Another approach is based on the straightforward re-moval of the SIE by using Perdew-Zunger correction.11

However, such SIE corrected functionals are inferior to their

parent functionals in terms of performance.12 For these rea-sons the correction of the SIE, while maintaining high accu-racy, remains a very important challenge in DFT.

In this Communication, we address the SIE problemwithin Kohn-Sham !KS" DFT.13 The adiabatic connection!AC" is modeled using exact exchange, GGAs and meta-GGAs, which gives a new exchange-correlation energy func-tional that is exact for one-electron densities. The functionalis accurate for thermochemistry and kinetics, while havingmany attractive formal properties. We will show that overallthe new functional, at similar computational cost, signifi-cantly improves upon commonly used functionals such asBLYP or B3LYP.

Our theoretical framework is the adiabatic connection,which has guided and illuminated the construction ofexchange-correlation functionals for many years.14–18 Con-sider a family of N-electron systems, Hv!""= T+ Vee!rN ,""+%i,#

N v#!ri ,"", where the electron-electron interaction andthe external potential are parametrized in terms of ", "! #0,1$. We focus on the linear adiabatic connection,Vee!rN ,""=%i$j" /rij, that links the KS system of noninter-acting particles !"=0" with the physical system !"=1". Theexchange-correlation energy is exactly defined by thecoupling-constant integration formula

Exc#!$ = &0

1

W"#!$d" , !2"

where W"#!$= '%"(Vee(%")!J#!$. Here J#!$ is the Coulombenergy, J#!$=1/2*!!r"!!r!" / (r!r!(drdr!, and %" is theunique antisymmetric ground state wave function of H", as-suming v-representability. Following Langreth and Perdew14

the electron density is fixed along the path, !!r"=!0!r"=!"!r""", such that %"!r" yields !!r" and minimizes theexpectation value 'T+"Vee). Once the integrand of the adia-batic connection is known, Exc is obtained from Eq. !2".

The challenge of the AC approach is that the exact W" isunknown. Thus, our main goal is to build accurate approxi-mations of W". In contrast to the usual density functionala"Both authors contributed equally to this work.

THE JOURNAL OF CHEMICAL PHYSICS 124, 091102 !2006"

0021-9606/2006/124"9!/091102/4/$23.00 © 2006 American Institute of Physics124, 091102-1

Downloaded 06 Mar 2006 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Wλ = a +bλ

1 + cλEXC = a +

b

c

“1− loge(1 + c)

c

”MCY

(Ernzerhof, CPL 263 499 (1996))

4 / 17

Functional Development ...

A new mixing of Hartree-Fock and local density-functional theories

Axe1 D. Becke Department of Chemistry, Queen’s University, Kingston, Ontario, Canada K7L 3N6

(Received 12 August 1992; accepted 8 October 1992)

Previous attempts to combine Hartree-Fock theory with local density-functional theory have been unsuccessful in applications to molecular bonding. We derive a new coupling of these two theories that maintains their simplicity and computational efficiency, and yet greatly improves their predictive power. Very encouraging results of tests on atomization energies, ionization potentials, and proton affinities are reported, and the potential for future development is discussed.

I. INTRODUCTION

There has long been interest in the problem of coupling Hartree-Fock theory with local density-functional approx- imations for dynamical correlation (see Ref. 1 and refer- ences therein). In doing so, one attempts to exploit the obvious strengths of each partner. Hartree-Fock theory provides an exact treatment of exchange at a cost that scales well with molecular size and is a practical compu- tational tool even for large chemical systems. Unfortu- nately, it suffers well-known and severe deficiencies in de- scribing chemical bonding and cannot be used in thermochemical applications, excepting isodesmic reac- tions, without further corrections for “correlation.” These post-Hartree-Fock corrections (i.e., Moller-Plesset per- turbation theory, configuration interaction, etc.) do not scale well with molecular size and are, at least currently, impractical in large systems.

On the other hand, local density-functional correlation approximations can be evaluated extremely easily and quickly (i.e., by numerical integration of functionals de- pending only on total electronic density) and appear to offer a convenient alternative to conventional post- Hartree+Fock technology. Straightforward addition of electron-gas correlation approximations to Hartree-Fock energies has not, however, proven successful in thermo- chemical tests,’ and more sophisticated approaches have consequently been proposed. Unfortunately, these involve multiconfiguration self-consistent field (MCSCF) (Refs. 2-5) or generalized valence-bond (GVB) (Ref. 6) refer- ence states and, hence, the beauty and practicality of the original “Hartree-Fock plus density-functionals” idea is lost.

In the present work, we propose a new mixing of HartreeFock theory and local density-functional theory that performs surprisingly well on the thermochemical tests of Pople’s Gaussian-l (Gl ) data base.7-9 This new approach is no more costly than its predecessors, but de- livers superior accuracy and holds great promise for fur- ther development.

II. THEORETICAL BACKGROUND

The analysis of the present work will be phrased in the language of the Kohn-Sham formulation of density- functional theory. lo Other formulations are also possible

but will not be considered here.” We should note at the outset that the term ‘LHartree-Fock” theory is not formally appropriate in the Kohn-Sham density-functional context, and that the term “exact exchange” is preferable. The dis- tinctions are subtle, however, and will be dealt with at appropriate points in the subsequent text.

In Kohn-Sham density-functional theory, we imagine a reference system of ultimate simplicity: a system of inde- pendent noninteracting electrons in a common, one-body potential Vx, yielding the same density as the “real” fully- interacting system. To be more specific, we imagine a set of independent reference orbitals pi satisfying the following independent-particle Schrbdinger equation:

-!V2+j+ VKS*i=ef*i (1)

with a local one-body potential V,, defined such that the noninteracting density

p= 5 l$il” (2) i

equals the density of the “real” system. Then, we express the total electronic energy of the real, fully-interacting sys- tem as

E total= T, + s

P Vnucd3r

pWp(r2) d3rld3r2 + Exe , (3)

r12

where To is the kinetic energy of the noninteracting refer- ence system, the second and third terms are the nuclear interaction energy and the classical Coulomb self-energy, respectively, and the last term E,, is the density-functional “exchange-correlation” energy.

Equation (3), in fact, defines the Kohn-Sham exchange-correlation energy. Although innocuous in ap- pearance, Exe contains, buried within it, all the details of two-body exchange and dynamical correlation and a kinetic-energy component as well. Nevertheless, it can be shown” that E,, depends uniquely on the total electronic density distribution and that the Kohn-Sham local poten- tial in Eq. ( 1) is given by

vKS = vn”c + vel + vXC 2 (4)

where Vel is the usual electronic Coulomb potential,

1372 J. Chem. Phys. 98 (2), 15 January 1993 0021-9606/93/021372-6$06.00 @ 1993 American Institute of Physics

Downloaded 27 Sep 2002 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

Wλ = a + bλ EXC = a +b

2H&H→ B3LYP

Self-interaction-free exchange-correlation functional for thermochemistryand kinetics

Paula Mori-Sánchez,a! Aron J. Cohen,a! and Weitao YangDepartment of Chemistry, Duke University, Durham, North Carolina 27708

!Received 20 January 2006; accepted 30 January 2006; published online 3 March 2006"

We develop a self-interaction-free exchange-correlation functional which is very accurate forthermochemistry and kinetics. This is achieved by theoretical construction of the functional formand nonlinear fitting. We define a simple interpolation of the adiabatic connection that uses exactexchange, generalized gradient approximation !GGA" and meta-GGA functionals. The performanceis optimized by fitting a small number of empirical parameters. Overall the new functional improvessignificantly upon hybrids and meta-GGAs while correctly describing one-electron systems. Themean absolute error on a large set of reaction barriers is reduced to 1.99 kcal/mol. © 2006American Institute of Physics. #DOI: 10.1063/1.2179072$

The theoretical formulation and implementation of im-proved exchange-correlation energy functionals is at theheart of density-functional theory !DFT" development.1 Im-pressive progress has been made from the early local densityapproximation !LDA"2 to generalized gradient approxima-tions !GGAs"3–5 and from GGAs to more recent implicitfunctionals constructed from the KS orbitals. Hybrid func-tionals which include a fraction of exact exchange6 andmeta-GGAs which make use of orbital kinetic energydensities,7 have led to the high accuracy and the wide usageof DFT. Despite their success, these functionals still exhibitimportant failures. For instance, B3LYP,5,6 the most popularof present functionals, gives errors greater than 4.0 kcal/molfor reaction energy barriers. Higher accuracy is needed.

Many of the remaining problems of current DFT ap-proximations are associated with the self-interaction error!SIE". An electron does not interact with itself. In a one-electron system, this leads to the requirement that the ex-change energy exactly cancels the Coulomb energy and thecorrelation energy vanishes,

Ex#!$ = ! J#!$; Ec#!$ = 0, !1"

where ! denotes any one-electron density. Availableexchange-correlation functionals violate one or both condi-tions, perhaps with the exception of Becke’s recent work.8

The self-interaction error is undoubtedly the origin ofmany qualitative and quantitative failures which plague DFT.Some of these are dissociation of one-electron systems andradicals, treatment of fractional number of electrons, chargetransfer processes, underestimation of reaction barriers, andover estimation of polarizabilities.9,10

There have been two main avenues for SIE correctionwhich have not been totally successful. One case is exempli-fied by hybrid functionals, which only partially reduce theSIE. Another approach is based on the straightforward re-moval of the SIE by using Perdew-Zunger correction.11

However, such SIE corrected functionals are inferior to their

parent functionals in terms of performance.12 For these rea-sons the correction of the SIE, while maintaining high accu-racy, remains a very important challenge in DFT.

In this Communication, we address the SIE problemwithin Kohn-Sham !KS" DFT.13 The adiabatic connection!AC" is modeled using exact exchange, GGAs and meta-GGAs, which gives a new exchange-correlation energy func-tional that is exact for one-electron densities. The functionalis accurate for thermochemistry and kinetics, while havingmany attractive formal properties. We will show that overallthe new functional, at similar computational cost, signifi-cantly improves upon commonly used functionals such asBLYP or B3LYP.

Our theoretical framework is the adiabatic connection,which has guided and illuminated the construction ofexchange-correlation functionals for many years.14–18 Con-sider a family of N-electron systems, Hv!""= T+ Vee!rN ,""+%i,#

N v#!ri ,"", where the electron-electron interaction andthe external potential are parametrized in terms of ", "! #0,1$. We focus on the linear adiabatic connection,Vee!rN ,""=%i$j" /rij, that links the KS system of noninter-acting particles !"=0" with the physical system !"=1". Theexchange-correlation energy is exactly defined by thecoupling-constant integration formula

Exc#!$ = &0

1

W"#!$d" , !2"

where W"#!$= '%"(Vee(%")!J#!$. Here J#!$ is the Coulombenergy, J#!$=1/2*!!r"!!r!" / (r!r!(drdr!, and %" is theunique antisymmetric ground state wave function of H", as-suming v-representability. Following Langreth and Perdew14

the electron density is fixed along the path, !!r"=!0!r"=!"!r""", such that %"!r" yields !!r" and minimizes theexpectation value 'T+"Vee). Once the integrand of the adia-batic connection is known, Exc is obtained from Eq. !2".

The challenge of the AC approach is that the exact W" isunknown. Thus, our main goal is to build accurate approxi-mations of W". In contrast to the usual density functionala"Both authors contributed equally to this work.

THE JOURNAL OF CHEMICAL PHYSICS 124, 091102 !2006"

0021-9606/2006/124"9!/091102/4/$23.00 © 2006 American Institute of Physics124, 091102-1

Downloaded 06 Mar 2006 to 129.234.4.10. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Wλ = a +bλ

1 + cλEXC = a +

b

c

“1− loge(1 + c)

c

”MCY

(Ernzerhof, CPL 263 499 (1996))

4 / 17

Wλ = a +bλ

1 + cλ

Choice of a, b, c ....

a = W0 = EX

b = ∂Wλ∂λ

˛λ=0

c chosen to reproduce BLYP Wλ at some λ

MCY1 (1e SI free)

5 / 17

Wλ = a +bλ

1 + cλ

Choice of a, b, c ....

a = W0 = EX

b = ∂Wλ∂λ

˛λ=0

c chosen to reproduce BLYP Wλ at some λ

MCY1 (1e SI free)

5 / 17

Wλ = a +bλ

1 + cλ

Choice of a, b, c ....

a = W0 = EX

b = ∂Wλ∂λ

˛λ=0

c chosen to reproduce BLYP Wλ at some λ

MCY1 (1e SI free)

5 / 17

PE curve of H2 in RKS – static correlation / fractional spins

2 4 6 83 5 7 9 10

FCI

E/

Har

tree

R / bohr

B3LYP

1

MCY1

HF

−0.8

−0.9

−1.0

−1.1

−1.2

Why poor? Mathematical form or approximate parameters?

How well does it perform when all three parameters are determined so as toreproduce exact properties of the adiabatic connection?

6 / 17

Strategy - determine accurate properties using Full CI

0

0

W0

EXC

−TC

λ

W1

1

Wλ

W FCI0 = EFCI

X = −1

2JFCI

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

∂λ

˛λ=0

= −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj(like MP2!)

... evaluated using ’exact’ KS orbitals [Wu & Yang, JCP 118 2498 (2003)]

Then choose a, b, c to reproduce these three quantities exactly.

7 / 17

Strategy - determine accurate properties using Full CI

0

0

W0

EXC

−TC

λ

W1

1

Wλ

W FCI0 = EFCI

X = −1

2JFCI

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

∂λ

˛λ=0

= −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj(like MP2!)

... evaluated using ’exact’ KS orbitals [Wu & Yang, JCP 118 2498 (2003)]

Then choose a, b, c to reproduce these three quantities exactly.

7 / 17

Strategy - determine accurate properties using Full CI

0

0

W0

EXC

−TC

λ

W1

1

Wλ

W FCI0 = EFCI

X = −1

2JFCI

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

∂λ

˛λ=0

= −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj(like MP2!)

... evaluated using ’exact’ KS orbitals [Wu & Yang, JCP 118 2498 (2003)]

Then choose a, b, c to reproduce these three quantities exactly.

7 / 17

Strategy - determine accurate properties using Full CI

0

0

W0

EXC

−TC

λ

W1

1

Wλ

W FCI0 = EFCI

X = −1

2JFCI

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

∂λ

˛λ=0

= −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj(like MP2!)

... evaluated using ’exact’ KS orbitals [Wu & Yang, JCP 118 2498 (2003)]

Then choose a, b, c to reproduce these three quantities exactly.

7 / 17

Strategy - determine accurate properties using Full CI

0

0

W0

EXC

−TC

λ

W1

1

Wλ

W FCI0 = EFCI

X = −1

2JFCI

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

∂λ

˛λ=0

= −1

2

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj(like MP2!)

... evaluated using ’exact’ KS orbitals [Wu & Yang, JCP 118 2498 (2003)]

Then choose a, b, c to reproduce these three quantities exactly.

7 / 17

EXC = EXC(a, b, c) = a + (b/c)“

1− loge (1+c)c

”

E = T FCIs + V FCI

ne + JFCI + Vnn + EXC

Discrepancy from FCI quantifies ability of AC form to describe H2 dissociation

Denote results AC1

8 / 17

EXC = EXC(a, b, c) = a + (b/c)“

1− loge (1+c)c

”

E = T FCIs + V FCI

ne + JFCI + Vnn + EXC

Discrepancy from FCI quantifies ability of AC form to describe H2 dissociation

Denote results AC1

8 / 17

2 4 6 83 5 7 9 10

FCI

E/

Har

tree

R / bohr

B3LYP

1

MCY1

HF

−0.8

−0.9

−1.0

−1.1

−1.2

2 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

−0.8

−0.9

−1.0

−1.1

−1.2

9 / 17

2 4 6 83 5 7 9 10

FCI

E/

Har

tree

R / bohr

B3LYP

1

MCY1

HF

−0.8

−0.9

−1.0

−1.1

−1.22 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

−0.8

−0.9

−1.0

−1.1

−1.2

9 / 17

AC1

λ

Wλ

MCY1

−0.65

−0.66

−0.67

−0.68

−0.69

−0.70

−0.71

−0.72

−0.73

−0.740.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

MCY1

−0.8

−0.9

−1.0

−1.1

−1.2

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

GL2 divergence!

10 / 17

AC1

λ

Wλ

MCY1

−0.65

−0.66

−0.67

−0.68

−0.69

−0.70

−0.71

−0.72

−0.73

−0.740.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

MCY1

−0.8

−0.9

−1.0

−1.1

−1.2

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

GL2 divergence!

10 / 17

AC1

λ

Wλ

MCY1

−0.65

−0.66

−0.67

−0.68

−0.69

−0.70

−0.71

−0.72

−0.73

−0.740.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

MCY1

−0.8

−0.9

−1.0

−1.1

−1.2

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

GL2 divergence!

10 / 17

AC1

λ

Wλ

MCY1

−0.65

−0.66

−0.67

−0.68

−0.69

−0.70

−0.71

−0.72

−0.73

−0.740.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 4 6 83 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr1

MCY1

−0.8

−0.9

−1.0

−1.1

−1.2

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

AC1

λ

Wλ

MCY1

−0.35

−0.40

−0.45

−0.50

−0.55

−0.60

−0.650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

GL2 divergence!

10 / 17

W AC1λ = a +

bλ

1 + cλ

W AC6λ = a + b exp(−cλ)

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC6FCI

E/

Har

tree

R / bohr

11 / 17

W AC1λ = a +

bλ

1 + cλ

W AC6λ = a + b exp(−cλ)

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC6FCI

E/

Har

tree

R / bohr

11 / 17

W AC1λ = a +

bλ

1 + cλ

W AC6λ = a + b exp(−cλ)

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC1FCI

E/

Har

tree

R / bohr

−0.90

−0.95

−1.00

−1.05

−1.10

−1.15

2 4 6 81 3 5 7 9 10

AC6FCI

E/

Har

tree

R / bohr

11 / 17

Other 2 electron systems? He-isoelectronic series!

0.02

0.01

0.03

0.04

0.05

0.06

−0.02

−0.01

0.00

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC2

AC4AC2(H&H)

AC4(H&H)

∆E

/H

artr

ee

B3LYP0.00

0.001

−0.001

−0.002

−0.003

−0.004

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC1AC3AC5AC6AC7AC8AC9

∆E

/H

artr

ee

12 / 17

Other 2 electron systems? He-isoelectronic series!

0.02

0.01

0.03

0.04

0.05

0.06

−0.02

−0.01

0.00

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC2

AC4AC2(H&H)

AC4(H&H)

∆E

/H

artr

ee

B3LYP

0.00

0.001

−0.001

−0.002

−0.003

−0.004

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC1AC3AC5AC6AC7AC8AC9

∆E

/H

artr

ee

12 / 17

Other 2 electron systems? He-isoelectronic series!

0.02

0.01

0.03

0.04

0.05

0.06

−0.02

−0.01

0.00

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC2

AC4AC2(H&H)

AC4(H&H)

∆E

/H

artr

ee

B3LYP0.00

0.001

−0.001

−0.002

−0.003

−0.004

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC1AC3AC5AC6AC7AC8AC9

∆E

/H

artr

ee

12 / 17

Why do most improve, but three get worse from H−to Ne8+?How do we achieve such high accuracy for large Z?

In the Z →∞ high-density limit, the exact AC is linear

Wλ = EX + 2EC,GL2λ (EXC = EX + EC,GL2)

As Z →∞, most of the AC forms exactly reproduce this.

But, three of the forms do not behave in this manner. They are the forms thatbecome less accurate from H−to Ne8+.

13 / 17

Why do most improve, but three get worse from H−to Ne8+?How do we achieve such high accuracy for large Z?

In the Z →∞ high-density limit, the exact AC is linear

Wλ = EX + 2EC,GL2λ (EXC = EX + EC,GL2)

As Z →∞, most of the AC forms exactly reproduce this.

But, three of the forms do not behave in this manner. They are the forms thatbecome less accurate from H−to Ne8+.

13 / 17

0.02

0.01

0.03

0.04

0.05

0.06

−0.02

−0.01

0.00

H− He Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Ne8+

AC2

AC4AC2(H&H)

AC4(H&H)

∆E

/H

artr

ee

B3LYP

14 / 17

0.60.2 0.80.40.00

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07

−0.08

−0.09

0 1

H−HeLi+

Be2+

B3+

C4+

N5+

O6+

F7+

Ne8+

WAC1

λ−

WAC1

0/

Har

tree

λ

15 / 17

Conclusions

J. Chem. Phys. 126 244104 (2007)J. Chem. Phys. 129 064105 (2008)

Illustration of how accurate Ψ can provide insight into DFT

Simple AC forms can be very accurate for 2e systems, when exact inputdata provided

Key role of GL2 energy in H2 (R →∞) and He-series (Z →∞)

16 / 17

Acknowledgments

Michael Peach

Andy Teale

17 / 17