david j tozer · density functional theory (dft) electronic structure method based on the...

DFT: Some Important Lessons from Atoms & their Ions

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

1 / 22

Density Functional Theory (DFT)

Electronic structure method based on the one-electron density, ρ(r)

Formally exact, approximate in practice; often good accuracy

Computationally simple

Extremely popular

E [ρ] = Ts [ρ] + Eext[ρ] + J[ρ] + EXC[ρ]

„−1

2∇2 + vext(r) + vJ(r) + vXC(r)

«ϕp(r) = εpϕp(r)

Aim: To illustrate how simple atoms/ions provide key insight ...

2 / 22

1. The Adiabatic Connection

Hλ = −1

∇2i + λVee +

vλ(ri )

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

Wλdλ

Wλ = 〈ψλ|Vee|ψλ〉 − J

3 / 22

1. The Adiabatic Connection

Hλ = −1

∇2i + λVee +

vλ(ri )

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

Wλdλ

Wλ = 〈ψλ|Vee|ψλ〉 − J

3 / 22

Exact Properties

Wλ Wλ = 〈ψλ|Vee|ψλ〉 − J

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

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

˛λ=0

= 2EC,GL2 = −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα + εβ − εi − εj− 2

〈φi |vX − vNLX |φα〉2

εα − εi

4 / 22

Exact Properties

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

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

˛λ=0

= 2EC,GL2 = −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα − εi

4 / 22

Exact Properties

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

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

˛λ=0

= 2EC,GL2 = −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα − εi

4 / 22

Exact Properties

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

W1 = 〈ψ1|Vee|ψ1〉 − J = Vee − J

∂Wλ

˛λ=0

= 2EC,GL2 = −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

εα − εi

4 / 22

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 approximations 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 computational tool even for large chemical systems. Unfortu- nately, it suffers well-known and severe deficiencies in describing 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 thermochemical 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) reference 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 further 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 independent 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 system as

E total= T, + s

P Vnucd3r

pWp(r2) d3rld3r2 + Exe , (3)

where To is the kinetic energy of the noninteracting reference 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 potential 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

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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

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"

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 +

“1− loge(1 + c)

”MCY

Q: How well can simple Wλ forms model the exact AC?A: Test using the two-electron He-isoelectronic series!

5 / 22

I. INTRODUCTION

p= 5 l$il” (2) i

E total= T, + s

P Vnucd3r

2H&H→ B3LYP

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

Exc#!$ = &0

W"#!$d" , !2"

Wλ = a +bλ

1 + cλEXC = a +

“1− loge(1 + c)

”MCY

5 / 22

I. INTRODUCTION

p= 5 l$il” (2) i

E total= T, + s

P Vnucd3r

2H&H→ B3LYP

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

Exc#!$ = &0

W"#!$d" , !2"

Wλ = a +bλ

1 + cλEXC = a +

“1− loge(1 + c)

”MCY

5 / 22

I. INTRODUCTION

p= 5 l$il” (2) i

E total= T, + s

P Vnucd3r

2H&H→ B3LYP

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

Exc#!$ = &0

W"#!$d" , !2"

Wλ = a +bλ

1 + cλEXC = a +

“1− loge(1 + c)

”MCY

5 / 22

Strategy - determine accurate properties using Full CI

W FCI0 = EFCI

X = −1

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

˛λ=0

= −1

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 define parameters (a, b, c) in Wλ forms to reproduce these 3 quantities.

6 / 22

W FCI0 = EFCI

X = −1

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

˛λ=0

= −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

6 / 22

W FCI0 = EFCI

X = −1

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

˛λ=0

= −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

6 / 22

W FCI0 = EFCI

X = −1

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

˛λ=0

= −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

6 / 22

W FCI0 = EFCI

X = −1

W FCI1 = V FCI

ee − JFCI

∂W FCIλ

˛λ=0

= −1

Xi,j,α,β

|(αi |βj)− (αj |βi)|2

6 / 22

EXC = EXC(a, b, c)

E = T FCIs + V FCI

ne + JFCI + Vnn + EXC

Discrepancy from FCI quantifies ability of AC form to describe He series

7 / 22

EXC = EXC(a, b, c)

E = T FCIs + V FCI

ne + JFCI + Vnn + EXC

Discrepancy from FCI quantifies ability of AC form to describe He series

7 / 22

W AC1λ = a +

1 + cλ

W AC2λ = a + bλ

W AC3λ = a + bλ+ cλ2

W AC4λ = a + b

1 + λ

«W AC5λ = a + b

1 + λ

W AC6λ = a + b exp(−cλ)

W AC7λ = a + b log(1 + cλ)

W AC8λ = a + b tanh(−cλ)

W AC9λ = a + bλ exp(−cλ)

7 / 22

−0.02

−0.01

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

AC4AC2(H&H)

AC4(H&H)

−0.001

−0.002

−0.003

−0.004

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

AC1AC3AC5AC6AC7AC8AC9

8 / 22

−0.02

−0.01

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

AC4AC2(H&H)

AC4(H&H)

B3LYP0.00

−0.001

−0.002

−0.003

−0.004

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

AC1AC3AC5AC6AC7AC8AC9

8 / 22

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+.

Linearity is not widely-known!

9 / 22

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+.

Linearity is not widely-known!

9 / 22

−0.02

−0.01

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

AC4AC2(H&H)

AC4(H&H)

10 / 22

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

H−HeLi+

11 / 22

2. vXC(r) and functional homogeneity

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

∆XC = (I− A) for open-shell systems

Q: Do local DFT functionals yield vXC(r) ≈ vN−δXC (r) + (I−A)

A: Test using the one-electron H isoelectronic series!

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

vN−δXC (r)

vN+δXC (r)

vN−δXC (r)

vN+δXC (r)

εHOMO = −I

vN−δXC (r) = −vJ(r) = − 1

r+ e−2Zr (Z + 1/r)

12 / 22

lower solid curve is vGPZ (r). These are compared with the

corresponding dashed unrestricted alpha BLYP potentials.

!The BLYP potentials were computed using a very largeeven-tempered basis set representation of the Hartree-Fock

density; Hartree-Fock energies with this basis set differed

from the exact !Z2/2 energies by a maximum of 10!4

Eh for

the four systems." The divergence of the BLYP potential atshort range is unimportant for basis set calculations.

For all four systems vGPZ (r) lies well below the BLYP

potential, despite the BLYP total energy being within 0.02Eh

of the exact value. The difference between vGPZ (r) and the

BLYP potential indicates that if vGPZ (r) is the exact potential,

then BLYP must give very poor potentials for hydrogenic

atoms, the error increasing with Z . Indeed, similar conclu-

sions would be drawn for many-electron systems #2$, al-though this seems nonintuitive given the general success of

these common functionals. The large discrepancy between

vGPZ (r) and BLYP is consistent with our observation that

fitting asymptotically vanishing potentials using simple ex-

pansions of the density and its gradient gives energies that

are too low #2$.Our investigations, together with the work of PPLB, sug-

gest that the exact exchange-correlation potential should lie

well above vGPZ (r). In a restricted formalism, the HOMO and

LUMO in hydrogenic atoms are degenerate, and the poten-

tials are predicted theoretically to differ by the hardness #8$(I!A)/2, where I and A are the ionization energy and elec-

tron affinity, respectively. For the four systems in Fig. 1 the

hardness increases with Z , taking the values "0.24Eh ,

"0.86Eh , "1.80Eh , and "2.42Eh , respectively. The upper

solid curve in Figs. 1!a"–1!d" represents the shifted potentialvGPZ (r)"(I!A)/2. For the energetically important regions,

this shifted potential agrees well with the BLYP potential for

all four systems, and this is particularly striking, considering

the large variation in the hardness. This agreement between

the potentials when the hardness is introduced, coupled with

the general success of the BLYP functional at describing

energetics, appears to support the theoretical work of PPLB.

!It is argued that one should use an unrestricted formalism indensity functional theory #9$ although our results suggestthat the alpha results are approximately equivalent in the two

formalisms." In this case the potential vGPZ (r)"(I!A)/2 is

close to the exact potential, and this is represented well by

the BLYP potential in energetically important regions. How-

ever, with increasing r , the shifted potential, which gives the

exact density, slowly approaches the hard-

FIG. 1. Potentials for the hydrogenic atoms !a" H, !b", Li2", !c" C5", and !d" O7" plotted as a function of radial distance. The dashed

curve is the BLYP potential. The lower solid curve is vGPZ (r), and the upper solid curve is vGP

Z (r)"(I!A)/2.

56 2727EXACT HYDROGENIC DENSITY FUNCTIONALS

vN−δXC

(r) +(I−A)

13 / 22

lower solid curve is vGPZ (r). These are compared with the

corresponding dashed unrestricted alpha BLYP potentials.

!The BLYP potentials were computed using a very largeeven-tempered basis set representation of the Hartree-Fock

density; Hartree-Fock energies with this basis set differed

from the exact !Z2/2 energies by a maximum of 10!4

Eh for

the four systems." The divergence of the BLYP potential atshort range is unimportant for basis set calculations.

For all four systems vGPZ (r) lies well below the BLYP

potential, despite the BLYP total energy being within 0.02Eh

of the exact value. The difference between vGPZ (r) and the

BLYP potential indicates that if vGPZ (r) is the exact potential,

then BLYP must give very poor potentials for hydrogenic

atoms, the error increasing with Z . Indeed, similar conclu-

sions would be drawn for many-electron systems #2$, al-though this seems nonintuitive given the general success of

these common functionals. The large discrepancy between

vGPZ (r) and BLYP is consistent with our observation that

fitting asymptotically vanishing potentials using simple ex-

pansions of the density and its gradient gives energies that

are too low #2$.Our investigations, together with the work of PPLB, sug-

gest that the exact exchange-correlation potential should lie

well above vGPZ (r). In a restricted formalism, the HOMO and

LUMO in hydrogenic atoms are degenerate, and the poten-

tials are predicted theoretically to differ by the hardness #8$(I!A)/2, where I and A are the ionization energy and elec-

tron affinity, respectively. For the four systems in Fig. 1 the

hardness increases with Z , taking the values "0.24Eh ,

"0.86Eh , "1.80Eh , and "2.42Eh , respectively. The upper

solid curve in Figs. 1!a"–1!d" represents the shifted potentialvGPZ (r)"(I!A)/2. For the energetically important regions,

this shifted potential agrees well with the BLYP potential for

all four systems, and this is particularly striking, considering

the large variation in the hardness. This agreement between

the potentials when the hardness is introduced, coupled with

the general success of the BLYP functional at describing

energetics, appears to support the theoretical work of PPLB.

!It is argued that one should use an unrestricted formalism indensity functional theory #9$ although our results suggestthat the alpha results are approximately equivalent in the two

formalisms." In this case the potential vGPZ (r)"(I!A)/2 is

close to the exact potential, and this is represented well by

the BLYP potential in energetically important regions. How-

ever, with increasing r , the shifted potential, which gives the

exact density, slowly approaches the hard-

FIG. 1. Potentials for the hydrogenic atoms !a" H, !b", Li2", !c" C5", and !d" O7" plotted as a function of radial distance. The dashed

curve is the BLYP potential. The lower solid curve is vGPZ (r), and the upper solid curve is vGP

Z (r)"(I!A)/2.

56 2727EXACT HYDROGENIC DENSITY FUNCTIONALS

vN−δXC

(r) +(I−A)

13 / 22

Fixing the asymptotic potential

A. The hydrogen atom

We commence by considering the effect of the

asymptotic exchange correction on the hydrogen atom,

where the exchange potential and orbital eigenvalues are

known exactly, to within a constant. Following an earlier

study,5 we use an extensive sp basis set, capable of describ-

ing both occupied and virtual orbitals. The exchange-only

Kohn–Sham equations have been solved to determine

HCTH-X and HCTH-X!ACX" potentials. The exact con-tinuum exchange potential is given by

vX,#exact!r"!"

r#e"2r! 1#

r"#C , !40"

where C is some constant. To allow comparison between the

exact and calculated potentials we set C to equal the calcu-

lated asymptotic value of the HCTH-X!ACX" potential of0.2345 au. $Note this is very close to the theoretical

asymptotic potential within a restricted formalism of (I

"A)/2!0.24 au.7% In Fig. 1 we compare this exact potentialwith the HCTH-X and HCTH-X!ACX" potentials. The

HCTH-X potential agrees well with the exact potential up to

about 2 au, but breaks down asymptotically. Apart from the

unphysical feature in the connection region, the HCTH-

X!ACX" potential is in excellent agreement with the exactpotential.

In Table I we present eigenvalues associated with these

potentials. The HCTH-X potential binds the 1s and 2s or-

bitals; all the remainder lie in the continuum. Apart from the

1s eigenvalue, all the HCTH-X!ACX" values are positive.Their associated orbitals are still bound, however, because

FIG. 1. The exact exchange potential vX,#exact(r) $Eq. !40"

with C!0.2345 au% for the hydrogen atom !dottedcurve", plotted as a function of radial distance, com-pared to self-consistent !a" HCTH-X and !b" HCTH-X!ACX" alpha spin potentials !solid curves". All quan-tities are in au.

3511J. Chem. Phys., Vol. 112, No. 8, 22 February 2000 Exchange potential

Downloaded 07 May 2001 to 129.234.4.1. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

εBLYP εBLYP(AC) εBLYP(AC) −1/(2n2)-0.2345

1s −0.2683 −0.2655 −0.5000 −0.50002s −0.0070 +0.1092 −0.1253 −0.12502p +0.0000 +0.1091 −0.1254 −0.1250...

......

5s +0.000 +0.2145 −0.0200 −0.02005p +0.000 +0.2145 −0.0200 −0.0200

14 / 22

vN−δXC (r) is the XC potential from EXC[ρ] = −J[ρ].

This functional is exact for one-electron systems, but poor for N > 1.

Now consider the alternative EXC[ρ] = − J[ρ]N

This is also exact for one-electron systems, and much better for N > 1.

Its potential is vN−δXC (r) + J

JN2 = 5Z

16 ≈(I−A)

15 / 22

Homogeneity of EXC[ρ]

EX[ρ] = CX

Zρ4/3(r)dr → 1

EX[ρ]

ZvX(r)ρ(r)dr =

Q: Is exact EXC[ρ] approximately homogeneous of some degree?A: Test using open-shell atoms H−Cl!

Following Zhao et al., define an ’effective homogeneity’ for the XC functional

keff =1

EXC[ρ]

ZvXC(r)ρ(r)dr

and evaluate using near-exact EXC[ρ] and vXC(r) (from ab initio densities).

16 / 22

Homogeneity of EXC[ρ]

EX[ρ] = CX

Zρ4/3(r)dr → 1

EX[ρ]

ZvX(r)ρ(r)dr =

Q: Is exact EXC[ρ] approximately homogeneous of some degree?A: Test using open-shell atoms H−Cl!

Following Zhao et al., define an ’effective homogeneity’ for the XC functional

keff =1

EXC[ρ]

ZvXC(r)ρ(r)dr

and evaluate using near-exact EXC[ρ] and vXC(r) (from ab initio densities).

16 / 22

keff =1

EXC[ρ]

ZvXC(r)ρ(r)dr

But we can use three different potentials!

vXC(r) = vN−δXC (r)

vXC(r) = vN+δXC (r) = vN−δ

XC (r) + (I − A)

vXC(r) = v averageXC (r) = vN−δ

XC (r) +(I − A)

kN−δeff =

EXC[ρ]

ZvN−δ

XC (r)ρ(r)dr

kN+δeff = kN−δ

eff +N(I − A)

EXC[ρ]

kaverageeff = kN−δ

eff +N(I − A)

2EXC[ρ]

17 / 22

sumption vxc(r)!vZMP(r). From Eq. !1", their calculationswere therefore implicitly on the electron-deficient side of the

integer. We generalize their analysis to examine the effective

homogeneity on both the electron-deficient and electron-

abundant sides of the integer. Substituting Eqs. !1" and !2"into Eq. !4" we obtain the open-shell result

keffN"#!

Exc$%& ! vZMP!r"%!r"dr, !5"

keffN##!

Exc$%& ! vZMP!r"%!r"dr#N!I"A "

Exc$%&. !6"

For a given system, these values indicate the effective homo-

geneity that the exact functional must exhibit if accurate en-

ergies and densities are to be obtained in association with the

potentials in Eqs. !1" and !2". Approximations to the inte-grals and exchange-correlation energies on the right-hand

side of Eqs. !5" and !6" have been computed by Morrisonand Zhao $8&, using CI densities, for the first- and second-row open-shell atoms. They do not present results for the H

atom, although since the ZMP potential for this atom is the

negative of the Hartree potential, and Exc$%& is the negativeof the Hartree energy, we trivially obtain keff

N"#!2.0. Valuesof (I"A) for all the atoms are presented in Ref. $9&. Wehave computed values of keff

N$# using these data, and present

the results in Fig. 1.

It is clear that both keffN"# and keff

N## are highly system de-

pendent. This indicates that for these atoms, the exact func-

tional cannot be well represented on the limiting electron-

deficient or electron-abundant sides of the integer by any

functional that is approximately homogeneous in the density.

Zhao et al. made an equivalent observation in their electron-

deficient study of the closed-shell atoms He, Be, Ne, and Ar

$1&. We observe that the value of keffN"# for the Si atom ap-

pears anomalous. While the value of keffN"# increases approxi-

mately linearly from B!C!N , the analogous behavior for

Al!Si!P is very different. We also note that Si is one of

the two atoms for which Parr and Ghosh $10& have presenteddifferent exchange-correlation energies to Morrison and

In Fig. 1 we also present values of the quantity

keffaverage! 1

2 $keffN###keff

N"#& , !7"

which represents the effective homogeneity of a functional

that gives the exact energy, and a potential for integer N that

averages over the exact discontinuity, i.e.,

vxc!r""average! 12 $vxc!r""N"##vxc!r""N##& !8"

vxc!r""average!vZMP!r"#!I"A "

2. !9"

The dotted line in Fig. 1 corresponds to keff!4/3. For theatoms H to F, the values of keff

average are reasonably close to

4/3, particularly for the heavier atoms. For the second-row

atoms, the values for Na, Al, P, S, and Cl are strikingly close

to 4/3. We also observe that if the value of keffN"# for Si is

lowered such that there is an approximately linear increase

from Al to P—as is the case for the first row—then the value

of keffaverage is also very close to 4/3. Further investigation is

required to resolve this discrepancy with Si. However, ex-

cluding this atom, the results of Fig. 1 indicate that for these

open-shell atoms, a functional whose potential averages over

the exact discontinuity must be approximately homogeneous

of degree 4/3 if it is to yield accurate energies. This is a

particularly good approximation for the heavier atoms.

IV. RELATION TO CONTINUUM FUNCTIONALS

Conventional local or GGA-type functionals such as LDA

$11&, BLYP $12&, and PW91 $13& are continuum functionals.Their potentials do not exhibit integer discontinuities, and it

has previously been argued that such functionals should

therefore give potentials that average over the discontinuity

in the exact potential $14–16&. !The situation is more com-plicated for functionals involving a fraction of exact orbital

exchange $17&." The results of the previous section supportthis requirement of an average behavior. Continuum func-

tionals are dominated by exchange, whose major contribu-

tion is the local density term Cx'%4/3(r)dr. This is exactlyhomogeneous of degree 4/3 in the density, and so continuum

functionals are also approximately homogeneous of degree

4/3. Figure 1 shows that for the open-shell atoms, any func-

tional that is approximately homogeneous of degree 4/3 can

only yield accurate energetics and densities if its potential

approximately averages over the discontinuity.

Our previous studies $7,18,19& on both closed- and open-shell systems also support the requirement of such average

behavior. In the general case, the potential that averages over

the discontinuity !in a restricted formalism" may be writtenas $7&

vxc!r""average!vZMP!r"#vxc( , vxc

( )!I"A "

2, !10"

FIG. 1. The effective homogeneities keffN$# $Eqs. !5" and !6"& and

keffaverage $Eq. !7"& for the first- and second-row open-shell atoms,

computed using the data in $8& and $9&.

PRA 58 3525EFFECTIVE HOMOGENEITY OF THE EXCHANGE- . . .

18 / 22

”Your paper is quite interesting. The average keff

leveling off at about 4/3 is remarkable, except for Si ...”

”Yes, I think leaving Si as questionable is appropriate”

and hopefully it will be resolved later”

19 / 22

”Your paper is quite interesting. The average keff

leveling off at about 4/3 is remarkable, except for Si ...”

”Yes, I think leaving Si as questionable is appropriate”

and hopefully it will be resolved later”

19 / 22

Orbital energies and negative electron affinities from density functionaltheory: Insight from the integer discontinuity

Andrew M. Teale,1,3 Frank De Proft,2 and David J. Tozer3,a!

1Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo,P.O. Box 1033, Blindern, N-0315 Oslo, Norway2Eenheid Algemene Chemie (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium3Department of Chemistry, University of Durham, South Road, Durham DH1 3LE, United Kingdom

!Received 7 May 2008; accepted 27 June 2008; published online 30 July 2008"

Orbital energies in Kohn–Sham density functional theory !DFT" are investigated, paying attentionto the role of the integer discontinuity in the exact exchange-correlation potential. A series ofclosed-shell molecules are considered, comprising some that vertically bind an excess electron andothers that do not. High-level ab initio electron densities are used to calculate accurate orbitalenergy differences, !", between the lowest unoccupied molecular orbital !LUMO" and the highestoccupied molecular orbital !HOMO", using the same potential for both. They are combined withaccurate vertical ionization potentials, I0, and electron affinities, A0, to determine accurate “average”orbital energies. These are the orbital energies associated with an exchange-correlation potential thataverages over a constant jump in the accurate potential, of magnitude !XC= !I0!A0"!!", as givenby the discontinuity analysis. Local functional HOMO energies are shown to be almost an order ofmagnitude closer to these average values than to !I0, with typical discrepancies of just 0.02 a.u. Forsystems that do not bind an excess electron, this level of agreement is only achieved when A0 is setequal to the negative experimental affinity from electron transmission spectroscopy !ETS"; itdegrades notably when the zero ground state affinity is instead used. Analogous observations aremade for the local functional LUMO energies, although the need to use the ETS affinities is lesspronounced for systems where the ETS values are very negative. The application of an asymptoticcorrection recovers the preference, leading to positive LUMO energies !but bound orbitals" for thesesystems, consistent with the behavior of the average energies. The asymptotically corrected LUMOenergies typically agree with the average values to within 0.02 a.u., comparable to that observedwith the HOMOs. The study provides numerical support for the view that local functionals exhibita near-average behavior based on a constant jump of magnitude !XC. It illustrates why a recentlyproposed DFT expression involving local functional frontier orbital energies and ionization potentialyields reasonable estimates of negative ETS affinities and is consistent with earlier work on thefailure of DFT for charge-transfer excited states. The near-average behavior of theexchange-correlation potential is explicitly illustrated for selected systems. The nature of hybridfunctional orbital energies is also mentioned, and the results of the study are discussed in terms ofthe variation in electronic energy as a function of electron number. The nature of DFT orbitalenergies is of great importance in chemistry; this study contributes to the understanding of thesequantities. © 2008 American Institute of Physics. #DOI: 10.1063/1.2961035$

I. INTRODUCTION AND BACKGROUND

Kohn–Sham density functional theory !DFT" is the mostwidely used electronic structure method. Practical calcula-tions rely on an accurate representation of the exchange-correlation energy and its functional derivative, theexchange-correlation potential vXC!r". Perdew et al.1 demon-strated that a plot of the exact total electronic energy as afunction of electron number comprises a series of straightline segments, with derivative discontinuities at the integers.This has the important consequence that the exact exchange-correlation potential jumps discontinuously as the electronnumber increases through an integer. The issue has attracted

significant recent interest.2–15 In particular, Sagvolden andPerdew3 re-examined the assumptions in the original proof.They concluded that the discontinuity is rigorously a con-stant jump for one-electron systems, and they established astrong presumption that it is a constant jump for many-electron systems. !A constant jump refers to a shift in thepotential by the same amount at all points in space;3 thevalue of the jump is both system and electron number depen-dent".

In the present study, we shall consider the implicationsof the integer discontinuity in practical DFT calculations us-ing local functionals such as generalized gradient approxima-tions !GGAs". We shall show that local functional frontierorbital energies are consistent with a potential that averagesover a constant jump, the magnitude of which is given by thediscontinuity analysis. The results will provide key insight

a"Author to whom correspondence should be addressed. FAX: #44 191 3844737. Electronic mail: [email protected].

Downloaded 30 Jul 2008 to 129.234.4.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

20 / 22

Conclusions

J. Chem. Phys. 129 064105Phys. Rev. A. 56 2726Phys. Rev. A. 58 3524

Have illustrated how calculations on small atoms and their ions can provide keyinsight into DFT, specifically:

The adiabatic connection in the high-Z (high-density) limit;

Shifts in the exchange-correlation potential;

Homogeneity of EXC[ρ].

21 / 22

Acknowledgments

Michael Peach

Andy Teale

22 / 22

david j tozer · density functional theory (dft) electronic structure method based on the...

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