F I J : A COLUMBUS Compatible program for finding

confluences

• Andy Young• At the Yarkony Group• JHU

USES OF FIJ AND POLYHES

• Polyhes -> minimization of arbitrary states• Polyhes -> minimization of the crossing

seam• FIJ -> Fitting potential surfaces• FIJ -> Mapping Loci of Confluences

Definition of a Confluence

2 seams with different # of dimensions meet; Confluence embedded in seam of higher

symmetry

Some molecules known to have confluences

• BH2

• O3

• C(3P) + H2

• HNCO• Possibly adding the CH2CHO radical soon

Why is a confluence important?

• Suppression of geometric phase effect• Polar angle in the gh plane is normally

arbitrary • Possible effect on dynamics

Criteria for the Existence of a Confluence

• Will first consider a general 2 by 2 symmetric Hamiltonian with G and –G as diagonal components and W as off diagonal.

• Will assume that W is a product of potentials associated with the two separate symmetries. This will be used to define a quantity that can identify a confluence.

Generalized Hamiltonian

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

S R G R W R G R W R

W R S R G R W R G R

⎛ ⎞ ⎛ ⎞− −→⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠

ur ur ur ur ur

ur ur ur ur ur

Allow

1 2( ) ( ) ( )W R V R V R=ur ur ur

then

1 2

1 2

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

G R V R V RG R W R

V R V R G RW R G R

⎛ ⎞⎛ ⎞ −−→ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

ur ur urur ur

ur ur urur ur

Defining the g-h plane for regular Hamiltonian

( ) ( )

( ) ( )

g R G R

h R W R

t g h

= ∇

= ∇

= ×

ur ur ur

r ur ur

r ur r

Recall also that• If we approximate

then G0=S • We removed common part

of diagonal components• If we are at a degeneracy,

then the zeroth part of an expansion of W is also zero.

• So all components of our generalized Hamiltonian starts with linear terms.

0 1( ) ( ) ( )G R G R G R≈ +ur ur ur

In terms of electronic wavefunctions Psi(I,J)

( )12 I I J Jg H H= Ψ ∇ Ψ − Ψ ∇ Ψ

ur

I Jh H= Ψ ∇ Ψr

With our ‘new’ Hamiltonian, define

( )1,21 2( ) ( ) ( )t G R V R V R= ∇ ×∇

uur ur ur ur

( ) ( )1 22 1( ) ( )V R g h V R g h= × + ×ur ur r ur ur r

0=

Hyper-cylindrical coordinates

{ }1 2, , , ,... mz z zρ θ

int 2m N= −

The seam coordinates are

{ }kz

Some “intermediate” or reduced quantities

( ) ( )2 2 2 2cos sinq g hθ θ= +

( )( )

sinarctan

coshg

θλ

θ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

Uses of the Polar Coordinates:First Order Energy in terms of the Tuning and Coupling Coordinates and the Tilt Parameters

( )

( ) ( )

( )

( )( )( ) )()sin()cos(

)(sin)(cos)sin()cos(

1

1

)(sin)(cos)sin(1)cos(121

21

2222

2222

θρθθρ

θθρθθρ

θθρθθρ

qssE

hgssE

hsh

s

gsg

s

hghsh

gsg

E

EHHs

yx

yx

IJIJy

IJIJx

IJIJIJIJ

IJJJIIIJ

±+=

+±+=

⋅=

⋅=

+±⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅=

∆=Ψ∇Ψ+Ψ∇Ψ=

r

r

rr

r

Decomposition of the Derivative Coupling

1i

mIJ IJ IJ IJ

zi

f f f fρ θ=

= + +∑

Expressions in terms of Hamiltonian elements expansion polynomials M

( ) ( )( )

( )( )

12

zIJ M Mf z

q q

ρ

θ

θ θλ θ ρ

θ θ θ θ θ⎛ ⎞∂ ∂ ∂

= + +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

0z =

( ) ( )( )

1 12

IJ Mf

q

ρ

θ

θλ θ

ρ ρ θ θ θ⎛ ⎞⎛ ⎞ ∂ ∂

= + ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

Maintaining the Position of the Seam Singularity Through Coordinate Transformation Requires

( ) ( ) ( )( )1, sin ,cosIJ IJ IJx yf f fθ θ θ

ρ⎛ ⎞

= −⎜ ⎟⎝ ⎠

The last equation on the previous slide shows that singularity can be isolated to a single term, and the remaining parts of the derivative coupling can be used to optimize seam coordinates, search for confluences, and characterize PES’s in the vicinity of the intersection.

Derivative coupling asymmetry in gh plane near a confluence

( ) ( ) ( ) ( )2 2 2 2 2

1 12 2 2 cos sin

gh ghq g h

λ θρ θ θ θ θ

∂= =

∂ +

0θ =for12hg

=

2πθ =for

12gh

=

~0 for h/g < 0.001

Search Method

1

m

kk kk

Z zχ ξ=

=∑ur r

( )2,k hk

ha

ξ = −

( ){ }2,hka

11

m

kk

χ=

=∑

Analytic values for low order expansion coefficients

( ) ( ) ( ) ( ) ( ) ( ) ( )1, 2,1 sin 2 cos cos sin2 k

h hIJIJ z k k

k

E q f a az

λ θ λ θ θρ⎛ ⎞∂

− ∆ + = +⎜ ⎟∂⎝ ⎠

Unit First-Order Cone = Aqua; Quadratic Cone = Red

Cs 12A’-22A’’

Nearby MEX’s of different symmetries; Cs

Nearby MEX’s of different symmetries; C1

References

• Yarkony, Matsika. J.Phys.Chem.A. vol.106, pg. 2580, (2002.)

• Yarkony. J.Phys.Chem.A. vol. 105, pg. 2642. (2001.)• Ivanic, Atchity,Ruedenberg. JCP vol. 107, pg. 4307.

(1997).• Yarkony, JCP vol. 112, pg. 2111. (2000.)• Yarkony, JCP, in press.• Young, Yarkony, JCP, in press.

• I would like to thank my advisor, Dr. Yarkony.

• I would also like to thank all of the COLUMBUS developers who have enabled us to produce results more efficiently.

Things that we think would be worthwhile with regards to

COLUMBUS• Reduce code size with fortran 90 modules.• Use select_kind values and allocated memory;

remove “work arrays” in favor of allocatable types or actual pointers (now officially supported in fortran language.)

• Change parallel CI from GA based to MPI based.• Fix parallel CI to run with cidrtfl from cidrtms.x

and not just from cidrt.x.• Calculate interstate density matrix without using

sum rules (to improve efficiency.)