( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

From Low-D Component Functions to D-reduction

( ) ( ) ( )

( ) ( ) ( )

2

1 2 1 2

1 2

(1) (2)

1 2 0

1 1

( )

, ,..., 12... 1 2

, ,...,

, ,... , ...

, ,..., ... , ,...,

D

nD

n n

n

CD

D i i ij i j

i i j

Cn D

i i i i i i D D

i i i

f x x x f f x f x x

f x x x f x x x

= < =

= + + + +

+ + +

∑ ∑

∑

Terms dependent on subsets of coordinates

( )

( ) ( )

( ) ( ) ( ) ( )

0

0

0

| ,

, , |

i i i

ij i j i j i i j j

f f y

f x f x y f

f x x f x x y f x f x f

=

= −

= − − −

…

Multimode: slices don't minimise global error, separate sample needed for each term

HDMR: one sample, but costly integrals

( ) ( ) ( )2

.D

HDMR

K

f f dµ − ∫ x x x

Using subsets of original coordinated leads to a combinatorial explosion of terms

Using subsets of original coordinated leads to a combinatorial explosion of terms

( ) ( ) ( )

( ) ( )

2 1,

0

,D

Dij

ij i j k k

kKk i j

i i j j

f x x w x f d

f x f x f

− =≠

=

− − −

∏∫ x x

No integrals: NN+HMDR, max order=f(sampling density): Manzhos, Carrington, J. Chem. Phys. 125, 084109 (2006)

No integrals: NN+HMDR, max order=f(sampling density): Manzhos, Carrington, J. Chem. Phys. 125, 084109 (2006)

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

Component Functions:Neural Networks

Represent a multi-D function with a small no. of simple 1D functions

●Math: universal approximator

●Chem: System/molecule-independent PES representation method in a dynamics-friendly form

f i

NN x1 , x2 , ... , x Dd=∑q=1

N

cq wqxdq

x =ex: f i

NN x=∑q=1

N

cq∏j=1

D

ew

qjx

jAny non-

linear function

Quadrature-friendly NN: Manzhos, Carrington, J. Chem. Phys. 125, 194105 (2006)Quadrature-friendly NN: Manzhos, Carrington, J. Chem. Phys. 125, 194105 (2006)

Fitting Algorithm

Algorithm: Manzhos, Yamashita, Carrington, Comput. Phys. Comm. 180, 2002 (2009) Algorithm: Manzhos, Yamashita, Carrington, Comput. Phys. Comm. 180, 2002 (2009)

# of y's < D = D-reduction

# of y's < D = D-reduction

Chemisorbed State

Intermolecular bonding weakened to favor dissociation

Free N2O

1621 cm-1

331 cm-1581 cm-1

997 cm-1

-21058

-21057.9

-21057.8

-21057.7

-21057.6

-21057.5

-21057.4

-21057.3

-21057.2

7.1 8.1 9.1 10.1

Ncx

E, e

V

Dissociation into N2+O

Chemisorbed state-0.53 eV

physisorbed state -0.23 eV

-21060

-21059.5

-21059

-21058.5

-21058

-21057.5

1 1.5 2 2.5 3 3.5

ON, A

E, eV

Barrier ~0.2 eVMin 9 coordsMin 9 coords

Unbiased Sampling in 15D

Rij

-1

●15 physically-motivated coordinates: asymptotics, permutational invariance

●Unbiased sampling of all configurations

●Much complicated fit

●15 physically-motivated coordinates: asymptotics, permutational invariance

●Unbiased sampling of all configurations

●Much complicated fit

D-reduction

D-reduction

2119

1178 1138

936 876 879 826 884

3504

0

1000

2000

3000

3 4 6 7 8 9 10 12 15

d

test set m

ae

, cm

-1

D-reduction allows for preservation of the density of sampling and ease of fit while using the best coordinates even as they expand D

D-reduction allows for preservation of the density of sampling and ease of fit while using the best coordinates even as they expand D

● 4,300 unique fitting data

● 1,845 unique test data

● 61,450 total symmetry augmented points

● 1.75 data per dimension

Manzhos, Yamashita, Surf. Sci. submittedManzhos, Yamashita, Surf. Sci. submitted

Dissociative Adsorption

100200

350500

7501000

50K

300K

600K

0

20

40

60

80

100

Pdiss, %

Ekin, meVT

50K

300K

600K

D-reduction of the Configuration Space

Brown et al., JCP 129, 064118 (2008)1,2,4-trifluorocyclooctane

Linear NN D-reductionN

2O/Cu(001)

Linear NN D-reductionN

2O/Cu(001)

Is D-reduction Useful? Or Why Linear Is Good

( ) ( ) ( )( ) ( )*,n ij mdx x V q x q x xΨ Ψ∫� � � � � � �

i j

1. Choice of neurons

( )1 2

1 1

1 1

, exp

np p

N d

n np p n

n p

dNw x

n

n p

f q q c w x b

c e

= =

= =

= + =

′=

∑ ∑

∑ ∏

( ) ( )*iconst x

i m i n idx x e xγ γ⋅

∫

2. Integration in redundant coordinates

dx�

is a d-form of exterior algebra:

1 2 ... ddx dx dx dx= ∧ ∧ ∧�

,

' ' '

1 2

1 1 1

...d d d

k k k k dk k

k k k

dx b dq b dq b dq= = =

= ∧ ∧ ∧ ∑ ∑ ∑

�

dx →∫�

Several d-dimensional integrals in q →�

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )* *,

i j

i j m i m j

j j

ij i j n i n j

dq dq q q

V q q q q

γ γ

γ γ

×

×∫ Manzhos, Carrington, J. Chem. Phys. 125, 194105 (2006)

Manzhos, Carrington, J. Chem. Phys. 125, 194105 (2006)

Conclusions● PES allow for speedy modelling of high-D systems -

molecules on surfaces/particles

● For D>6, the configuration space is sparsely sampled (2-3 data/D) even with ~104-5 data = no PES exists for most systems

● Using NN-based low-D functions provides universality, simplicity, and tractability

● D-reduction increases the effective density of sampling w/out neglecting degrees of freedom

Thanks JSPS

for funding

Thanks JSPS

for funding