
• 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

Axel D. Becke and Erin R. Johnson *
Email: axel.becke@dal.ca
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) <
( Position , rather than time, dependent)

In Hartree-Fock theory , the total energy of many-electron system is given by:
E
HF
 
1
2 i

,

 i

 2
 i
 d
3 r
  
V nuc d
3 r

1
2


( r
1
)

( r
2
) d
3 r
1 d
3 r
2

E
X r
12
  

 



 i
N
  

1
2 i

E
X

E
X


E
X

E
X

 
1
2
 ij
 i

( r
1
)
 j

( r
1
)
 i

( r
2
)
 j

( r
2
) d
3 r
1 d
3 r
2 r
12
The “exchange” energy

The (

-spin) exchange energy can be rewritten as follows: where
E
X


1
2
 

( r
1
) h
X

( r
1
, r
2
) d r
12
3 r
2 d
3 r
1 h
X

( r
1
, r
2
)
 


1
( r
1
)
 ij
 i

( r
1
)
 j

( r
1
)
 i

( r
2
)
 j

( r
2
) is called the “exchange hole”.
Physical interpretation: each electron interacts with a “hole” whose shape
(in terms of r
2
) depends on the electron’s position r
1
.
When an electron is at r
1
, the hole measures the depletion of probability , with respect to the total electron density, of finding another electron of the same spin at r
2
. This arises from exchange antisymmetry.

It is simple to prove that…
• The hole is always negative .
• The probability of finding another same-spin electron at r
2 top” of the reference electron) is completely extinguished:
= r
1
(“on h
X

( r
1
, r
1
)
  

( r
1
)
Pauli or exchange “repulsion”!
• The hole always contains exactly (minus) one electron:
 h
X

( r , ) d
1 r
2
3 r
2
 
1

An electron plus its exchange hole always has zero total charge but in general a non-zero dipole moment ! This r
1
-dependent dipole moment is easily obtained by integrating over r
2
: d
X

( r
1
)






1
( r
1
)
 ij r ij

 i

( r
1
)
 j

( r
1
)



 r
1 r ij

  r
2
 i

( r
2
)
 j

( r
2
) d
3 r
2
Note that only occupied orbitals are involved.
(This reduces, for the H atom, to the exact dipole moment of the
H atom when the electron is at r
1
.)

In a spherical atom, consider the following simplified “2-point” picture: d
X
 e ( r ,
W
) h

( r - d
X

,
W
)
Notice that this picture generates higher multipole moments as well
(with respect to the nucleus as origin) given by
Nucleus


 
[ r
 
( r
 d
X

)

] and that all these moments depend only on the magnitude of the exchange-hole dipole moment.

This is significant because the magnitude d
X

( 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 2 nd -order GGA: d
X

( r )
 b 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)].

Therefore, the entire van der Waals theory that follows has two variants:
XX performs slightly better in rare-gas systems.
BR performs better in intermolecular complexes.
All our current work employs the BR (DFT) variant.

r
A
The Dispersion Interaction: Spherical Atoms
V int
( r
A
, r
B
) r
B
V int
( r
A
, r
B
) = multipole moments of electron+hole at r
A interacting with multipole moments of electron+hole at r
B

2 nd -Order Ground-State Perturbation Theory in the
Closure (Ünsold) Approximation
If the first -order, ground-state energy correction arising from a perturbation V pert is zero:
E
( 1 ) 
V pert

0
Then the second -order correction is approximately given by
E
( 2 )  

V pert
2

E avg
 where the expectation values are in the ground state and

E avg is the average excitation energy .

To evaluate the expectation value < V int
2 > square the multipole-multipole interaction V int
( r
A
, r
B
):
V int
2 ( r
A
, r
B
) = ( dipole-dipole + dipole-quadrupole + dipole-octopole + quadrupole-quadrupole
+…) 2
Then integrate the squared interaction over all r
A and r
B
.
This is a “semiclassical” calculation of < V int
2 > .

The result is E disp
 
C
6
R
6

C
8
R
8

C
10
R
10
   where C
6

2
 
1
2
3

A


E avg

1
2 
B
C
8

 
1
2 
A
 
2
2
  
B

E avg

2
2 
A
 
1
2 
B
C
10

4
3
 
1
2 
A
 
3
2 
B


E
 avg

3
2 
A
 
1
2 
B

14
5
 
2
2 
A


E avg

2
2 
B with atomic moment integrals given by
 

2  



( r )[ r
 
( r
 d
X

)

]
2 d
3 r


E avg

E avg
 
E
A
 
E

E

2
 
1
2
3

 where α is its dipole polarizability . This easily follows from the same
“semiclassical” 2 nd -order perturbation theory applied to the polarizability of each atom.
Thus our C
6
, C
8
, C
10
’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:
C
6
C
8
C
10
3.4 %
21.5 %
21.5 %

• Partition a molecular system into “atoms” using Hirshfeld weight functions: w i
( r )

 n
 at i
 n
( r at
)
( r ) where
 at is a spherical free atomic density placed at the appropriate nucleus and the n summation is over all nuclei.
w i
( r ) has value close to 1 at points near nucleus i and close to 0 elsewhere.
Also,
 i w i
( r )

1
• Assume that an inter molecular C m inter atomic C m ,ij
:
C m

( m = 6,8,10) can be written as a sum of
B A  i j
C m , ij i in A j in B

In the previous expressions for C m replace A and B with i and j :
C
6 , ij

2

3

1
2
(

E i
 i



1
2

E j
)
 j
C
8 , ij

 
1
2
 i
 
2
2

(

E i j


 

E j
)
2
2
 i
 
1
2
 j
C
10 , ij

4
3
 
1
2  i
 
3
2  j
(

E i


 
3
2

E j
)
 i
 
1
2  j 
14
5
 
2
2
(

E i
 i



2
2

E j
)
 j with
 

2  i generalized to
 

2
 i
 
 w i
( r )


( r )[ r
 
( r
 d
X

)

]
2 d
3 r

and, 
E i

2
 
3
 i
1
2  i where
 is the effective polarizability of atom i in A . We propose that i
 i


 r
3 r
3
 i ,
 i free
 i , free
 r
3  i
  r
3 w i
( r )

( r ) d
3 r
 r
3  i , free
  r
3
 i , free
( r ) d
3 r
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
 

2 are with respect to the position of nucleus i .
 i

Test set: H
2 and N
2 with He, Ne, Ar, Kr, and Xe.
Cl
2 with He, Ne, Ar, Kr, and Xe (except C
10
).
H
2
-H
2
H
2
-N
2
N
2
-N
2
(only H
2
-H
2 for C
10
).
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 C
6
16.5% for C
8
11.9% for C
10
(On a much more extensive test set of 178 intermolecular C
6
’s, the model has a MAPE of 9.1%)

Everything, so far, has been about the asymptotic dispersion series between atom pairs,
E disp
 
C
6
R
6

C
8
R
8

C
10
R
10
  
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…

Since we can compute C
6
, C
8
, and C
10 non-empirically, we can obtain non-empirical range information.
There is a “critical”
R c,ij where the three dispersion terms are approximately equal:
Take R c,ij as the average of , , and
The asymptotic dispersion series is obviously meaningless inside R c,ij
Therefore …
.

… we use
R c,ij to damp the dispersion energy at small internuclear separations as follows:
E disp
  i

 j
R
6 vdW
C
6 , ij

, ij
R ij
6

R
8 vdW
C
8 , ij

, ij
R ij
8

R
10 vdW
C
10 , ij

, ij
R ij
10
R vdW , ij
 a
1
R c , ij
 a
2 where R vdw,ij is an effective van der Waals separation with only two universal fit parameters.
Best-fit a
1 and a
2 values depend on the exchange-correlation theories with which the above is combined.
… which brings us to the second part of the talk
…

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

Axel D. Becke and Felix O. Kannemann
Email: axel.becke@dal.ca
Department of Chemistry, Dalhousie University,
Halifax, Nova Scotia, Canada

Exchange GGA Functionals B86, B86b, B88
E
X
B 86 
E
X
LDA 
0 .
0036



1



4 / 3


2
0 .
004

2


E
X
B 86 b 
E
X
LDA 
0 .
0036



4

/ 3

2


1

0 .
006

2

4

/ 5
E
X
B 88 
E
X
LDA 
0 .
0042



1


0 .
0252
4


/

3

2
 arcsin h


 where the local (spin) density part is E
X
LDA  
3
2 and the “reduced” (spin) density gradient is
3
4

 1 / 3



4

/ 3  
4

/ 3




 


4

/ 3

Exchange GGA Functionals PW86, PW91, PBE(96)
E
X
  e
X
LDA
(

) g
X
( s ) where e
X
LDA
(

)
 
3
4
3
(

)
1 / 3

4 / 3 g
X
PW 86
( s )


1

1 .
296 s
2 
14 s
4 
0 .
2 s
6
1

/ 15 g
X
PW 91
( s )

1

0 .
19645 s sinh
1

0 .

1
( 7 .
7956 s )

[ 0 .
2743

0 .
1508 exp(

100 s
2
)] s
2
19645 s sinh

1
( 7 .
7956 s )

0 .
004 s
4 g
X
PBE
( s )

1

0 .
804

1

0 .
804
0 .
21951 s
2
/ 0 .
804
The “reduced” (spin) density gradient is and, for spin polarized systems, s

 
2 ( 3

2
)
1 / 3

4 / 3
E
X
(


,


)

1
2

E
X
( 2


)

E
X
( 2


)


revPBE (Yang group) g
X
PBE
( s )

1

0 .
804

1

0 .
804
0 .
21951 s
2
/ 0 .
804 g
X revPBE
( s )

1

1 .
245

1

1 .
245
0 .
21951 s
2
/ 1 .
245

2
Plot is from Kannemann and Becke, JCTC 5, 719 (2009)

Ne
2

anything
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) g
X
PW 86 ( s )

E
X
  e
X
LDA
(

) g
X
( s )

1

1 .
296 s 2 
14 s 4 
0 .
2 s 6
1

/ 15 s

 
2 ( 3

2
)
1 / 3

4 / 3
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 

Use the non-empirical
“PBE” correlation functional:
Perdew, Burke, and Ernzerhof, PRL 77, 3865 (1996)
Therefore we have,
E
XC

E
PW
X
86 
E
C
PBE  XDM
E disp
XDM
E disp
 
1
2 i

 j
6
R vdW
C
6 , ij

, ij
R ij
6

8
R vdW
C
8 , ij

, ij
R ij
8

10
R vdW
C
10 , ij

, ij
R ij
10
R vdW , ij
 a
1
R c , ij
 a
2 with a
1 and a
2 to be determined.

a
1
a
2
2
2
2
(reference data from Tang and Toennies, JCP 118, 4976 (2003))
At the CBS limit, we find the best-fit values a
1
=0.65 a
2
=1.68
(Kannemann and Becke, to be published)

Gaussian 09, aug-cc-pV5Z, counterpoise, ultrafine grid, BEs in microHartree a
1
= 0.65 a
2
= 1.68
Å
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)

Axel D. Becke and Felix O. Kannemann
Email: axel.becke@dal.ca
Department of Chemistry, Dalhousie University,
Halifax, Nova Scotia, Canada

The “S22” and “S66” vdW Benchmark Sets of Hobza et al
S22
S66 hydrogen bonding dispersion other noncovalent interactions

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-XDM no CP (0.30)
CP (0.28)

S66
PW86PBE-XDM no CP (0.27)
CP (0.23)

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

• 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!

Axel D. Becke and Felix O. Kannemann
Email: axel.becke@dal.ca
Department of Chemistry, Dalhousie University,
Halifax, Nova Scotia, Canada

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

10
0
-10
-20
-30
PW86PBE+XDM
M06-2X
-40
-50
0.90
1.00
1.10
M05-2X
M06L
1.25
distance multiplier
1.50
2.00

25
20
15
10
5
0
-5
-10
-15
-20
0.90
1.00
1.10
PW86PBE+XDM
M06-2X-D3
M05-2X-D3
M06L-D3
1.25
distance multiplier
1.50
2.00

25
20
15
10
5
0
-5
-10
-15
-20
0.90
PW86PBE+XDM
PBE-D3
BLYP-D3
B97-D3
1.00
1.10
1.25
distance multiplier
1.50
2.00

10
5
0
-5
-10
-15
-20
0.90
1.00
1.10
PW86PBE+XDM
LC-ωPBE-D3
B3LYP-D3
B2-PLYP-D3
1.25
distance multiplier
1.50
2.00

Mean Absolute Percent Errors (MAPEs) versus geometry

60
50
40
30
80
70
20
10
0
0.90
1.00
1.10
PW86PBE+XDM
M06-2X
M05-2X
M06L
1.25
distance multiplier
1.50
2.00

25
20
15
10
35
30
5
0
0.90
1.00
1.10
PW86PBE+XDM
M06-2X-D3
M05-2X-D3
M06L-D3
1.25
distance multiplier
1.50
2.00

35
30
25
20
15
10
5
0
0.90
1.00
1.10
PW86PBE+XDM
PBE-D3
BLYP-D3
B97-D3
1.25
distance multiplier
1.50
2.00

35
30
25
20
15
10
5
0
0.90
1.00
1.10
PW86PBE+XDM
LC-ωPBE-D3
B3LYP-D3
B2-PLYP-D3
1.25
distance multiplier
1.50
2.00

(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 functional E
XC

E
X
GGA 
E
C
PBE  XDM
E disp 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)