A Density-Functional Theory for Covalent and Noncovalent Chemistry Review of the XDM (exchange-hole dipole moment) dispersion model of Becke and Johnson.

A Density-Functional Theory for Covalent and Noncovalent

Chemistry

• Review of the XDM (exchange-hole dipole moment) dispersion model of Becke and Johnson

• Combining the model with non-empirical exchange-correlation GGAs (Kannemann and Becke)

• Tests on standard bio-organic benchmark sets

Non-empirical and fast

Dispersion Interactions from the Exchange-Hole Dipole Moment:

The “XDM” model

Axel D. Becke and Erin R. Johnson*

Email: [email protected]

Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada

(*now at University of California, Merced)

What is the source of the “instantaneous” multipole moments that generate the dispersion interaction?

Suggestion:

Becke and Johnson, J. Chem. Phys. 122, 154104 (2005)

> Becke and Johnson, J. Chem. Phys. 127, 154108 (2007) <

The dipole moment of the exchange hole!(Position, rather than time, dependent)

In Hartree-Fock theory, the total energy of many-electron system is given by:

Xnuci

iiHF Eddr

dVdE 23

13

12

213

,

32 )()(

2

1

2

1rr

rrrr

N

ii

1

2

23

13

12

2211 )()()()(

2

1rr

rrrrdd

rE

ij

jijiX

XXX EEE

The “exchange” energy

)()()()()(

1),( 2211

121 rrrr

rrr

jij

ijiXh

13

23

12

211

),()(

2

1rr

rrr dd

r

hE X

X

The Exchange Hole

The ( -spin) exchange energy can be rewritten as follows:

where

is called the “exchange hole”.

Physical interpretation: each electron interacts with a “hole” whose shape (in terms of r2) depends on the electron’s position r1.

When an electron is at r1, the hole measures the depletion of probability, with respect to the total electron density, of finding another electron of the same spin at r2. This arises from exchange antisymmetry.

• The hole is always negative.

• The probability of finding another same-spin electron at r2=r1 (“on top” of the reference electron) is completely extinguished:

)(),( 111 rrr Xh

Pauli or exchange “repulsion”!

• The hole always contains exactly (minus) one electron:

1),( 23

21 rrr dhX

It is simple to prove that…

1111

1 )()()(

1)( rrrr

rrd ijX

ijji

23

222 )()( rrrrrij dji

Note that only occupied orbitals are involved.

An electron plus its exchange hole always has zero total charge but in general a non-zero dipole moment! This r1-dependent dipole moment is easily obtained by integrating over r2:

(This reduces, for the H atom, to the exact dipole moment of the H atom when the electron is at r1.)

Dispersion Model: the Basic Idea

h (r - dX, )

e- (r, )

Nucleus

dX

])([ Xdrr

In a spherical atom, consider the following simplified “2-point” picture:

Notice that this picture generates higher multipole moments as well (with respect to the

nucleus as origin) given by

and that all these moments depend only on the magnitude of the exchange-hole dipole moment.

This is significant because the magnitude dX(r) of the exchange-hole dipole moment can be approximated using local densities and the Becke-Roussel exchange-hole model [Phys. Rev. A 39, 3761 (1989)], a 2nd-order GGA:

where b is the displacement from the reference point of the mean position of the BR model hole [Becke and Johnson, J. Chem. Phys. 123, 154101 (2005)].

bd X )(r

Therefore, the entire van der Waals theory that follows has two variants:

Orbital (XX) based, or Density-functional (BR) based

XX performs slightly better in rare-gas systems.

BR performs better in intermolecular complexes.

All our current work employs the BR (DFT) variant.

rA

rBVint(rA,rB)

Vint(rA,rB) = multipole moments of electron+hole at rA interacting with

multipole moments of electron+hole at rB

The Dispersion Interaction: Spherical Atoms

2nd-Order Ground-State Perturbation Theory in the Closure (Ünsold) Approximation

If the first-order, ground-state energy correction arising from a perturbation Vpert is zero:

Then the second-order correction is approximately given by

avg

pert

E

VE

2

)2(

0)1( pertVE

where the expectation values are in the ground state and

is the average excitation energy. avgE

To evaluate the expectation value <Vint2>

square the multipole-multipole interaction Vint(rA,rB):

Vint2(rA,rB) = (dipole-dipole + dipole-quadrupole +

dipole-octopole + quadrupole-quadrupole +…)2

Then integrate the squared interaction over all rA and rB.

This is a “semiclassical” calculation of <Vint2>.

The result is 1010

88

66

R

C

R

C

R

CEdisp

avg

BA

avg

BABA

EEC

22

22

21

23

23

21

10 5

14

3

4

avg

BABA

EC

2

122

22

21

8

avg

BA

EC

2

12

16 3

2 where

rr 322 ])()[( ddrr X

with atomic moment integrals given by

What about ? avgE

BAavg EEE Assume that and that, for each atom,

3

2 21

E

where α is its dipole polarizability. This easily follows from the same “semiclassical” 2nd-order perturbation theory applied to the polarizability of each atom.

Thus our C6, C8, C10’s depend on atomic polarizabilities and moment integrations from Hartree-Fock (or KS) calculations!

No fitted parameters or explicitly correlated wavefunctions!

How well does it work?

On the 21 pairs of the atoms H, He, Ne, Ar, Kr, Xe,

the mean absolute percent errors are:

C6 3.4 %

C8 21.5 %

C10 21.5 %

From Free Atoms to Atoms in Molecules

• Partition a molecular system into “atoms” using Hirshfeld weight functions:

n

atn

ati

iw)(

)()(

r

rr

where at is a spherical free atomic density placed at the appropriate nucleus and the n summation is over all nuclei.

)(riw has value close to 1 at points near nucleus i and close to 0 elsewhere. Also,

i

iw 1)(r

• Assume that an intermolecular Cm (m=6,8,10) can be written as a sum of interatomic Cm,ij :

A

i

B

jijCC ,mm

i in A j in B

In the previous expressions for Cm replace A and B with i and j :

)(3

22

12

1,6

ji

jiij EE

C

)(

21

22

22

21

,8ji

jijiij EE

C

)(5

14

)(3

422

22

21

23

23

21

,10ji

ji

ji

jijiij EEEE

C

with i 2 generalized to

rrr 322 ])()[()( ddrrw Xii

and,

i

iiE

3

2 21

where i is the effective polarizability of atom i in A. We propose that

freeifreei

ii r

r,

,3

3

rrr 333 )()( dwrr ii rr 3,

3,

3 )( drr freeifreei

These are effective volume integrations. This is motivated by the well known qualitative (if not quantitative) general relationship between polarizability and volume. See Kannemann and Becke, JCP 136, 034109 (2012).

All radii r in the above integrals and in the integrals for are with respect to the position of nucleus i.

i 2

Test set: H2 and N2 with He, Ne, Ar, Kr, and Xe.

Cl2 with He, Ne, Ar, Kr, and Xe (except C10).

H2-H2 H2-N2 N2-N2 (only H2-H2 for C10).

Fully-numerical Hartree-Fock calculations on the monomers using the NUMOL program (Becke and Dickson, 1989).

MAPEs with respect to dispersion coefficients from frequency dependent MBPT polarizabilities:

12.7% for C6 16.5% for C8 11.9% for C10

(On a much more extensive test set of 178 intermolecular C6’s, the model has a MAPE of 9.1%)

Atom-Molecule and Molecule-Molecule Dispersion Coefficients

Everything, so far, has been about the asymptotic dispersion series between atom pairs,

The asymptotic series needs to be damped in order to avoid divergences when R is small.

i.e. need information about characteristic R values inside of which the asymptotic series is no longer valid.

The usual approach is to use empirical vdW radii.

However…

1010

88

66

R

C

R

C

R

CEdisp

“Critical” Interatomic Separation

Since we can compute C6, C8, and C10 non-empirically, we can obtain non-empirical range information.

There is a “critical” Rc,ij where the three dispersion terms are approximately equal:

Take Rc,ij as the average of , , and

The asymptotic dispersion series is obviously meaningless inside Rc,ij. Therefore …

… we use Rc,ij to damp the dispersion energy at small internuclear separations as follows:

where Rvdw,ij is an effective van der Waals separation with only two universal fit parameters. Best-fit a1 and a2 values depend on the exchange-correlation theories with which the above is combined.

… which brings us to the second part of the talk …

General Dispersion Energy Formula

2,1, aRaR ijcijvdW

ji ijijvdW

ij

ijijvdW

ij

ijijvdW

ijdisp RR

C

RR

C

RR

CE

1010,

,10

88,

,8

66,

,6

A. D. Becke and E. R. Johnson, J. Chem. Phys. 127, 154108 (2007)

Dispersion coefficients

E. R. Johnson and A. D. Becke, J. Chem. Phys. 124, 174104 (2006)

Damping functions

A. D. Becke and E. R. Johnson, J. Chem. Phys. 123, 154101 (2005)

Transform to a DFT

Key XDM Publications

Exchange-Correlation GGAs

XDM Dispersion

Axel D. Becke and Felix O. Kannemann

Email: [email protected]

Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada

+

It would be nice if we could use

“standard” XC-GGAs (instead of, eg., Hartree-Fock)

plus the XDM dispersion model

to treat vdW interactions.

Let’s look at the exchange part first.

2

23/486

004.010036.0LDA

XBX EE

5/42

23/486

006.010036.0LDA

XbB

X EE

harcsin0252.010042.0

23/488 LDA

XBX EE

Exchange GGA Functionals B86, B86b, B88

where the local (spin) density part is

3/43/4

3/1

4

3

2

3

LDAXE

and the “reduced” (spin) density gradient is 3/4

Exchange GGA Functionals PW86, PW91, PBE(96)

The “reduced” (spin) density gradient is

)()( sgeE XLDAXX where

3/43/1)3

(4

3)(

LDA

Xe

15/164286 2.014296.11)( ssssg PWX

41

22191

004.0)7956.7(sinh19645.01

)]100exp(1508.02743.0[)7956.7(sinh19645.01)(

sss

sssssg PW

X

804.0/21951.01

804.0804.01)(

2ssg PBE

X

3/43/12 )3(2

s

and, for spin polarized systems, )2()2(2

1),( XXX EEE

revPBE (Yang group)

804.0/21951.01

804.0804.01)(

2ssg PBE

X

245.1/21951.01

245.1245.11)(

2ssg revPBE

X

Now plot the exchange-only interaction energy in Ne2

Note that Hartree-Fock (exact exchange) is repulsive!

Plot is from Kannemann and Becke, JCTC 5, 719 (2009)

Ne2

Can get anything!

from artifactual binding to massive over-repulsion,

depending on the choice of functional!

Which exchange GGA best reproduces exact Hartree-Fock?

Lacks and Gordon, PRA 47, 4681 (1993)

Kannemann and Becke, JCTC 5, 719 (2009)

Murray, Lee, and Langreth, JCTC 5, 2754 (2009)

PW86, followed by B86b

Exchange Enhancement Factor [Zhang, Pan, Yang, JCP 107, 7921 (1997)]

PW86 is a completely non-empirical exchange functional!

Its 4 parameters are fit to a theoretical exchange-hole model.

Perdew and Wang, PRB 33, 8800 (1986)

)()( sgeE XLDAXX

15/164286 2.014296.11)( ssssg PWX

3/43/12 )3(2

s

That PW86 accurately reproduces Hartree-Fock repulsion energies is remarkable. The underlying theoretical model (truncated “GEA” hole) knows nothing about closed-shell atomic or molecular interactions! Could be a fortuitous accident?!

Nevertheless, no parameters need to be fit to data

ji ijijvdW

ij

ijijvdW

ij

ijijvdW

ijXDMdisp RR

C

RR

C

RR

CE

1010,

,10

88,

,8

66,

,6

2

1

2,1, aRaR ijcijvdW

XDMdisp

PBEC

PWXXC EEEE 86

Now, what about (dynamical) correlation?

Use the non-empirical “PBE” correlation functional:

Perdew, Burke, and Ernzerhof, PRL 77, 3865 (1996)

Therefore we have,

with a1 and a2 to be determined.

How to determine a1 and a2?

Fit to the binding energies of the prototypical dispersion-bound rare-gas systems

He2 HeNe HeAr Ne2 NeAr Ar2

(reference data from Tang and Toennies, JCP 118, 4976 (2003))

At the CBS limit, we find the best-fit values

a1=0.65 a2=1.68

(Kannemann and Becke, to be published)

Gaussian 09, aug-cc-pV5Z, counterpoise, ultrafine grid, BEs in microHartree

a1 = 0.65 a2 = 1.68Å

RMS%E = 4.2%

Note that in all subsequent benchmarking there is no (re)fitting of parameters!

Our functional is, from here on, essentially nonempirical in all its parts.

F.O. Kannemann and A.D. Becke, JCTC 5, 719 (2009)

Rare-gas diatomics (numerical post-LDA)

F.O. Kannemann and A. D. Becke, JCTC 6, 1081 (2010)

Intermolecular complexes (numerical post-LDA)

A.D. Becke, A.A. Arabi and F.O.Kannemann, Can. J. Chem. 88, 1057 (2010)

Dunning aDZ and aTZ basis-set calculations (post-G09)

GGA + XDM Publications

Tests on Standard BioOrganic Benchmark Sets

Axel D. Becke and Felix O. Kannemann

Email: [email protected]

Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada

S22

S66

hydrogen bondingdispersion

other noncovalent interactions

The “S22” and “S66” vdW Benchmark Sets of Hobza et al

References

S22: Jurecka, Sponer, Cerny, Hobza, PCCP 8, 1985 (2006)

S66(x8): Rezac, Riley, Hobza JCTC 7, 2427 (2011)

The next slides contain Mean Absolute Deviations (MADs) for the S22 and the S66 benchmark sets in comparison with other popular DFT methods. Data for all methods other than ours are from

Goerigk, Kruse & Grimme, CPC 12, 3421 (2011)

All computations employ the def2-QZVP basis set

PW86PBE-XDMno CP (0.30)CP (0.28)

S66

PW86PBE-XDMno CP (0.27)CP (0.23)

Aside

Can we do better by combining XDM with hybrid functionals?

(rather than the pure GGA, PW86+PBE)

Burns, Vazquez-Mayagoitia, Sumpter, Sherrill, JCP 134, 084107 (2011) S22 MAD (kcal/mol) for B3LYP-XDM and other DFT methods

However

• The pure GGA, PW86+PBE, is completely nonempirical (B3LYP, with 3 fitted parameters, is not)

• Density-fit basis sets can speed up pure GGA calculations, with no loss of accuracy, by an order of magnitude!

Tests on the S66x8 Set

(Nonequilibrium Geometries)

Axel D. Becke and Felix O. Kannemann

Email: [email protected]

Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada

S66x8

A benchmark set of 66 vdW complexes of bio-organic interest, at 8 intermonomer separations:

0.90, 0.95, 1.00, 1.05, 1.10, 1.25, 1.50, 2.00

(relative to the equilibrium intermonomer separation)

Rezac, Riley, Hobza JCTC 7, 2427 (2011)

Important because complex systems and materials may contain many vdW interactions between groups at nonequilibrium

(especially stretched) geometries!

Mean Percent Errors (MPEs)

versus geometry

0.90 1.00 1.10 1.25 1.50 2.00-50

-40

-30

-20

-10

0

10

distance multiplier

MP

E [

%]

PW86PBE+XDM

M06-2X

M05-2X

M06L

0.90 1.00 1.10 1.25 1.50 2.00-20

-15

-10

-5

0

5

10

15

20

25

distance multiplier

MP

E [

%]

PW86PBE+XDM

M06-2X-D3

M05-2X-D3

M06L-D3

0.90 1.00 1.10 1.25 1.50 2.00-20

-15

-10

-5

0

5

10

15

20

25

distance multiplier

MP

E [

%]

PW86PBE+XDM

PBE-D3

BLYP-D3

B97-D3

0.90 1.00 1.10 1.25 1.50 2.00-20

-15

-10

-5

0

5

10

distance multiplier

MP

E [

%]

PW86PBE+XDM

LC-ωPBE-D3

B3LYP-D3

B2-PLYP-D3

Mean Absolute Percent Errors (MAPEs)

versus geometry

0.90 1.00 1.10 1.25 1.50 2.000

10

20

30

40

50

60

70

80

distance multiplier

MA

PE

[%]

PW86PBE+XDM

M06-2X

M05-2X

M06L

0.90 1.00 1.10 1.25 1.50 2.000

5

10

15

20

25

30

35

distance multiplier

MA

PE

[%]

PW86PBE+XDM

M06-2X-D3

M05-2X-D3

M06L-D3

0.90 1.00 1.10 1.25 1.50 2.000

5

10

15

20

25

30

35

distance multiplier

MA

PE

[%]

PW86PBE+XDM

PBE-D3

BLYP-D3

B97-D3

0.90 1.00 1.10 1.25 1.50 2.000

5

10

15

20

25

30

35

distance multiplier

MA

PE

[%]

PW86PBE+XDM

LC-ωPBE-D3

B3LYP-D3

B2-PLYP-D3

pdf file of all S66x8 curves

(Notice the importance of the dispersion term!)

Interaction types in S66x8:

• H-bonding 23

• Dispersion 23

• Mixed (“other”) 20

Dispersion is further divided into:

pi-pi (10), aliphatic-aliphatic (5), and pi-aliphatic (8)

How good is PW86+PBE+XDM for ordinary thermochemistry?

Consider the functionalXDMdisp

PBEC

GGAXXC EEEE

with a variety of standard exchange GGAs in the first term.

On the “G3/99” benchmark set of 222 atomization energies of organic/inorganic molecules (Curtiss, Raghavachari, Pople) we obtain the following error statistics, in kcal/mol:

For standard hybrid functionals (eg., B3LYP) the MAE is of order 5-6 kcal/mol. The very best DFTs have MAE as small as 2-3 kcal/mol, but with fitted params!

Availability of XDM code

• Has been implemented (B3LYP-XDM) in Q-Chem by

Kong, Gan, Proynov, Freindorf, and Furlani, PRA 79, 042510 (2009)

• A “post-Gaussian09” code will be available from us by the end of 2012. Uses G09 to perform the PW86+PBE part, then adds XDM perturbatively. Can do Berny geometry optimizations using the EXTERNAL keyword! Very fast with density-fit basis sets!

Many thanks to:

Natural Sciences and Engineering Research Council of Canada

the Killam Trust of Dalhousie University (Killam Chair)

ACEnet (the Atlantic Computational Excellence Network)