A Density-Functional Theory for Covalent and Noncovalent Chemistry

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A Density-Functional Theory for Covalent and
Noncovalent Chemistry
Non-empirical and fast ?

• 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

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Dispersion Interactions from the Exchange-Hole
Dipole Moment The XDM model

• Axel D. Becke and Erin R. Johnson
• Email axel.becke_at_dal.ca
• Department of Chemistry, Dalhousie University,
  Halifax, Nova Scotia, Canada

(now at University of California, Merced)
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What is the source of the instantaneous
multipole moments that generate the dispersion
interaction? Suggestion Becke and Johnson, J.
Chem. Phys. 122, 154104 (2005) gt Becke and
Johnson, J. Chem. Phys. 127, 154108 (2007)
lt The dipole moment of the exchange hole!
(Position, rather than time, dependent)
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In Hartree-Fock theory, the total energy of
many-electron system is given by
The exchange energy
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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 electrons 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.
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It is simple to prove that

• The hole is always negative.
• The probability of finding another same-spin
  electron at r2r1 (on top of the reference
  electron) is completely extinguished

Pauli or exchange repulsion! The hole
always contains exactly (minus) one electron
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Dispersion Model the Basic Idea
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
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 r1.)
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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.
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This is significant because the magnitude dXs(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).
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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.
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The Dispersion Interaction Spherical Atoms
rA
Vint(rA,rB)
rB
Vint(rA,rB) multipole moments of
electronhole at rA interacting with
multipole moments of electronhole at rB
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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
where the expectation values are in the ground
state and is the
average excitation energy.
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To evaluate the expectation value ltVint2gt
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 ltVint2gt.
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The result is
where
with atomic moment integrals given by
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What about ?
Assume that
and that, for each atom,
where a 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,
C10s depend on atomic polarizabilities and
moment integrations from Hartree-Fock (or KS)
calculations! No fitted parameters or explicitly
correlated wavefunctions!
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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
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From Free Atoms to Atoms in Molecules
Partition a molecular system into atoms
using Hirshfeld weight functions
where
is a spherical free atomic density placed at the
appropriate nucleus and the n summation is over
all nuclei.
has value close to 1 at points near nucleus i and
close to 0 elsewhere. Also,
Assume that an intermolecular Cm (m6,8,10)
can be written as a sum of interatomic Cm,ij
i in A j in B
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In the previous expressions for Cm replace A and
B with i and j
with
generalized to
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and,
where
is the effective polarizability of atom i in A.
We propose that
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.
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Atom-Molecule and Molecule-Molecule Dispersion
Coefficients
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 C6s, the model has a MAPE of 9.1)
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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
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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
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General Dispersion Energy Formula
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
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Key XDM Publications
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
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Exchange-Correlation GGAs XDM Dispersion

Axel D. Becke and Felix O. Kannemann Email
axel.becke_at_dal.ca Department of Chemistry,
Dalhousie University, Halifax, Nova Scotia, Canada
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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. Lets look at the exchange part
first.
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Exchange GGA Functionals B86, B86b, B88
where the local (spin) density part is
and the reduced (spin) density gradient is
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Exchange GGA Functionals PW86, PW91, PBE(96)
where
The reduced (spin) density gradient is
and, for spin polarized systems,
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revPBE (Yang group)
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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)
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Ne2
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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
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Exchange Enhancement Factor Zhang, Pan, Yang,
JCP 107, 7921 (1997)
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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)
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 ?
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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.

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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 a10.65 a21.68 (Kannemann and
Becke, to be published)
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Gaussian 09, aug-cc-pV5Z, counterpoise, ultrafine
grid, BEs in microHartree
a1 0.65 a2 1.68Å RMSE 4.2
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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.
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GGA XDM Publications
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)
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Tests on Standard BioOrganic Benchmark Sets
Axel D. Becke and Felix O. Kannemann Email
axel.becke_at_dal.ca Department of Chemistry,
Dalhousie University, Halifax, Nova Scotia, Canada
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The S22 and S66 vdW Benchmark Sets of Hobza
et al
S22
S66
hydrogen bonding dispersion other noncovalent
interactions
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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
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PW86PBE-XDM no CP (0.30) CP (0.28)
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S66
PW86PBE-XDM no CP (0.27) CP (0.23)
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Aside Can we do better by combining XDM with
hybrid functionals? (rather than the pure GGA,
PW86PBE)
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Burns, Vazquez-Mayagoitia, Sumpter, Sherrill,
JCP 134, 084107 (2011) S22 MAD (kcal/mol) for
B3LYP-XDM and other DFT methods
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• However
• The pure GGA, PW86PBE, 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!

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Tests on the S66x8 Set (Nonequilibrium Geometries)
Axel D. Becke and Felix O. Kannemann Email
axel.becke_at_dal.ca Department of Chemistry,
Dalhousie University, Halifax, Nova Scotia, Canada
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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!
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Mean Percent Errors (MPEs) versus geometry
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Mean Absolute Percent Errors (MAPEs) versus
geometry
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pdf file of all S66x8 curves (Notice the
importance of the dispersion term!)

• Interaction types in S66x8
• H-bonding
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• Dispersion
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• Mixed (other)
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Dispersion is further divided into pi-pi (10),
aliphatic-aliphatic (5), and pi-aliphatic (8)
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How good is PW86PBEXDM for ordinary
thermochemistry? Consider the functional
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!
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• 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
  PW86PBE part, then adds XDM perturbatively. Can
  do Berny geometry optimizations using the
  EXTERNAL keyword! Very fast with density-fit
  basis sets!

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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)