Dissociative Recombination of LiH

2

+ + e-

DVR (Discrete Variable Representation) hodgepodge

PLUS!

Informal AMO theory seminar, May 7 2007 Dan Haxton, Greene Group

The Basics of Dissociative Recombination

e- + AB+

(-) (+)

A + B

E = I.E.(AB) E = D.E.(AB)

Electronic excitation energy is turned into nuclear kinetic energy.

>

Studies of Dissociative Recombinationin the Greene Group

DR is relevant to interstellar chemistry (diffuse ionized gas) – we have looked at● H

3

+ 1

● HCO+ 2

● LiH+ 3

● And isotopomers 4

1Kokoouline and Greene, PRA 68 102703 (2003) 2Mikhailov, Kokoouline, Larson, Tonzani, and Greene, PRA 74 032707 (2006) 3Curik and Greene, PRL 98 173201 (2007) 4Kokoouline and Greene, PRA 72 022712 (2005)

Direct versus Indirect DR

DIRECTCoupling through valence

state

r(A-B)

E

E0KE

Direct versus Indirect DR

INDIRECTCoupling through rydbergs

r(A-B)

E

E0KE

LiH2

+ : Both Direct and Indirect DR

may play a role

... and the two mechanisms may be intertwined

Li+

H23.6a

0

1.4a0

Measurements of R. D. Thomas, M. Larsson, et al.1 and E. Bahati, C. R. Vane et al.2 (submitted) indicate that all products are observed, in appreciable magnitude!

1 Albanova Univ., Sweden 2 ORNL

+ e- Li + H2, LiH + H, Li + H + H

77% 6% 17%

E (

eV)

rHH

(a0)

Candidate for a Direct DR channel

5.4 rHHLi

H

H

Excited state neutral:Li + H

2* (3∑

u)

~Ground stateLi + H

2

+ curve

E (

eV)

rHH

(a0)

Candidate for a Direct DR channel

5.4 rHHLi

H

H

E0 (v = 0 H

2)

Crossing at ~2a0!

Near classical turningpoint of v = 2 H

2

Results: Single partial wave

Results: Single partial wave

More partial waves: different electronicsymmetries couple differently to DR

pz

px

s-wave

HH

Li

zx

Enforcing outgoing wave boundaryconditions:

Siegert states and Exterior Complex Scaling

We use Siegert states and ECS to representthe outgoing waves corresponding to the

dissociating nuclei in DR.

e-

e-

[ This diagram represents: incoming waves in electronic channels; outgoing waves in all channels. ]

Enforcing outgoing wave boundary conditionsoutgoing wave boundary conditions is not trivialnot trivial with a computer.

In contrast, enforcing a zerozero boundary condition is easy.

Siegert statesSiegert states and ECSECS allow one to enforce anoutgoing wave boundary condition, and can beused with a wide variety of primitive basis sets.

Siegert States

Siegert states are defined within a BOXBOX, e.g., a radial interval from zero to some r

0.

They satisfy the time-independent Schrodinger equation inside the box, and obey a boundary

condition that depends on the eigenvalue:

Siegert States

Re(k)

Im(k

)

Spectrum in complex k plane

Figure from Santra, Shainline, and Greene PRA 71, 032703 (2005)

Antibound

BoundOutgoingIncoming

Tolstikhin et al [PRA 58, 2077 (1998)]expressed these equations as a

doubled-dimension generalized eigenvalue problem:

Hij = <φ

i|H|φ

j> L

ij = φ

i(r

0) φ

j(r

0)

Ψ(r) =∑i c

i φ

i(r)

Exterior Complex Scaling

R0

θ

Re(R)

Im(R)

R(r)

Real part of outgoing wave w/ECS

No ECS : box states

Represent the wavefunction for complex-valued coordinates. Zero boundary condition becomes

outgoing wave boundary condition.

What can I do with ECS or Siegertstates?

● Construct outgoing, incoming, or artificial- singularity-free principal value Green's function ● Create an absorbing boundary

[Santra, Shainline, and Greene PRA 71 032703 (2005)]

● Probably other stuff too

Siegert States Versus ECS

Siegert ECS ● Strange completeness relations (inner product)● Explicit expressions for Green's functions

● Full set (incoming, outgoing, bound, antibound) req'd for (formal) completeness ● If is nonzero, must (WKB?) correct wavefunction derivative at boundary

● Only outgoing, or only incoming, are complete ● For most accurate treatment, must analytically continue potential to complex r

DVR / FBR!The DVR (Discrete Variable Representation) andFBR (Finite Variable Representation) are methods ofevaluating potential energy matrix elements,

< φi| V |φ

j >

● A set of weights {wi}● A basis {φ

i}

Their ingredients are ● A set of points {x

i}

Generalized DVR / FBR refs: Light, Hamilton, and Lill, JCP 82 1400 (1985);Corey and Tromp, JCP 103 1812 (1995); Corey and Lemoine, JCP 97 4115 (1992);Szalay, JCP 105 6940 (1996)

DVR / FBR!

V |φi> = ∑

j=1...N v

ij |φ

j>

These techniques assume that the potential does notoperate beyond the (truncated) basis set, i.e. that

To the degree to which this is not the case, thesemethods make an additional error beyond the basisset truncation error, called the aliasing error. As a result, they are not variational.

DVR / FBR!These methods employ a transformation betweenfunction space (represented by coefficients {c

i}) and

point space (values of at points {xi}): Ψ(x

i) s

i )

Where is the matrix of basis function values φi(x

j) m

ij

{si}T = {c

i}

DVRIn a DVR based upon Gaussian quadrature1, there is a unitary transformation upon the basis {φ

i} to a

DVR basis {ϕi} in which each function corresponds to a

gridpoint and is zero on every other gridpoint.

< ϕi| V |ϕ

j > = δ

ij V(x

i)

This makes doing calculations in the DVR basis a very good idea, because the potential is diagonal.

1Dickinson and Certain, JCP 49 4209 (1968)

FBRThere are many variations of the FBR, which is performedin the basis {φ

i} -- e.g., more points than weights2. I use3

< φi| V |φ

j > =

{V(xi)}

Extensions to more than one dimension are problematic;the matrix is singular with equal # of points andbasis functions, and more gridpoints than basis functionssacrifices properties useful in, e.g., electronic structure.

2Bramley et al., JCP 100, 6175 (1994)3Czako, Szalay, and Csaszar, JCP 124 014110 (2006)

1 of 14, l =2

11 of 38, l =410 of 14, l =2

1 of 38, l =4

Views of Lebedev DVR functions

Views of Lebedev DVR functions

l = 3

l = 7 before iteration

l = 7

Lebedev DVR

Distorted 3D harmonic oscillator test results, Lebedev basis with l=4

N-Body problem in Relative Coordinates

For a system of N interacting particles, interacting via pairwisepotentials, the center-of-mass Hamiltonian may be written

Where the index i labels the mass-weighted interparticle spacings r

αβ , α being the particle index, 1 to N.

This is a differential equation in N(N-1)/2 variables.

should commute with and . . .

r

12 + r

23 + r

31 = 0

N-Body problem in Relative Coordinates

The simplest system one can apply this idea to is threeparticles in 1D.

r31

r12

r23

r31

r12

r23

N-Body problem in Relative Coordinates

The idea is to discretize this equation using a basisset, as usual. However, I'll discretize it in a basis ofproducts of functions of the relative coordinates,

and use an unphysical inner product, integrating over the interparticle distances,

N-Body problem in Relative Coordinates

Because there are M = N(N-1)/2 interparticle distances,the resulting basis set size is massive; the basisscales as A = n

phys

N nunphys

N(N-1)/2 - N , where nphys

is the

number of basis functions required for the physicaldegrees of freedom, and n

unphys is the number

required for the unphysical dofs. Ideally, nunphys

does

not exceed 5-10....

However, the potential is a one particle operator, andone may operate with the hamiltonian using M2n2 + A operations, and the basis is orthogonal.

N-Body problem in Relative Coordinates

However, the method allows the use of basis functionsin the relative coordinates (a la Hylleras) without lineardependence issues or involving a generalized eigenvalue problem.

It scales better than a non-DVR calculation in physical coordinates, which requires two-particle operators and

therefore scales as Nn2 + Mn4.

It may be useful, therefore, in problems with singularinterparticle interactions, e.g. Coulomb problems, forwhich DVR encounters difficulty.