Progress in Calculating the Potential Energy Surface of H3

+

Ludwik Adamowicz and Michele Pavanello, Department of Chemistry and Biochemistry

February 9th, 2012

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Goal: Accurate PES of H3+

MotivationInterstellar chemistry: (Hn)+

Spectroscopy: H3+

What has been done in the past?

Why Molecules with Hydrogen?

Method Single Point (cm-1) PES (cm-1)CI, (CC), (MPn) 200 – 10 200 – 10

R12 – CI, (CC, MPn) <10 <10

ECGs* < 10-3 0.02Our work ? ?

*Cencek JCP, 108, 2831 (1998) ; Explicitly Correlated Gaussians (ECGs)

What are ECGs?

Expansion in terms of basis functions

The basis set is made of explicitly correlated Gaussians with floating centers

rrrr MMM gcgcgc 2211

...exp

...exp,...,,2

23232

1212

222

21121

rr

rrrrrgkk

kknek

...exp

...exp,...,,2

23232

1212

2

222

2

11121

rr

SrSrrrrgkk

kkkknek

linear and non-linear parameters

The case of H3+

ner

r

r

2

1

r

Atoms

Molecules

jiij rrr

The cusp condition1. Electron-Nucleous cusp2. Electron-Electron cusp

Zr

Rr2

1

11

Zr

FRr

2

11)(

11

r

Kato’s condition*

*T. Kato (1957)

Cusp function:

The derivative of Ψ in this point counts!

Is the nucleous really a point charge?

Optimization of ФM

EnergyE

GradientG

Determine the step size and

move

PPP

gPPHggPHPggPHgP

gPHgccE

lklklk

M

i

M

jlklk

ˆˆˆ

;ˆˆˆˆˆˆˆˆˆ

;ˆˆ1 1

M

i

M

jlk

ilki

i

ji

kijnei

ki

gPHgx

ccxG

x

1 1

ˆˆ

,

The step size is determined as a function of G and E.

• Variational PrincipleMM

MM HE

|rel.non

Does our approach work?Single point calculation

Basis size CPU Time (days)

Energy1

(au)Energy2

(au)150 8 -1.343 835 599 83 -1.343 835 540300 40 -1.343 835 623 08 -1.343 835 615600 8 -1.343 835 624 94 -1.343 835 624

1000 0.1 -1.343 835 625 02 NA

1) Pavanello et. al. J. Chem. Phys., 130, 034104 (2009) 2) Cencek et al. Chem. Phys. Lett. 246, 417-420 (1995)

1. Non-linearity: M*7 parameters2. Encounter linear dependencies

5 or 6 electrons maximumI. Antisymmetrize electrons: ne! II. Basis set size: M2

Schrödinger EquationI. Born-Oppenheimer approximationII. RelativityIII. Coulomb HamiltonianImplementation – Parallelization – Numerical InstabilityI. Encounter linear dependenciesII. Memory constraints

Limits

What if we move the geometry?Can we carry out PES calculations?

1. Re-optimize from scratch the basis set for each PES grid point. a. Takes a long time to optimize the basis setb. Hundreds, sometimes thousands of geometries need to be considered

2. Guess the basis set from nearby geometriesa. How?b. Is it precise?c. Is the precision maintained for each grid point?

We need a benchmark!

Test of the spring method Benchmark PES of H3

+ Generated a 377-point PESThe wavefunction at a certain

geometry was generated from one of a nearby geometry

Pavanello et al. J. Chem. Phys. 130, 001033 (2009)

The spring model

Convergence dictated by the value of the analytical gradient ( GTG < 10-11 a.u. ) and not of

the energy

M=9006300 parameters

1st benchmark: D3h symmetry

-6 -4 -2 0 2 4 6

-1.36

-1.34

-1.32

-1.3

-1.28

-1.26

-1.24

-1.22

-1.2

-1.18 -6 -4 -2 0 2 4 6

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

Total Energy (a.u.) ΔE(a.u. x 10-8)

We notice:Our energies are always 0.01 cm-1 below the

best in the literature.Stretched geometries seem to show better

improvement

The more negative the bett

er!

2nd benchmark: C2v & asymmetric

-5 -4 -3 -2 -1 0 1 2 3

-10

-9.5

-9

-8.5

-8

-7.5

-7

-4 -3 -2 -1 0 1 2 3

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

asymmetric

C2v 3C2v 4

0 1 2 3 4 5 6 7 8

-12

-10

-8

-6

-4

-2

0

The challenge: a complete PES of H3+

toward sub 0.01 cm-1 accuracy

0 2 4 6 8 10 12

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

11A1

1E1

Tota

l Ene

rgy

(a.u

.)

ρ (a.u.)

R12

R 13

R23

Viegas, Alijah and Varandas, JCP (2007)Johnson, JCP (1980)Whitten and Smith (1968)

sinsin13

3

4sinsin1

3

3

4sinsin1

3

22

13

2223

22

12

R

R

R

3hD0sin

Alijah at al. usedMR-CI with cc-pV5Z

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

MR-CI vs "exact"1 2 3 4 5 6 7 8

-1.4

-1.35

-1.3

-1.25

-1.2

-1.15

-1.1

-1.05

-1Rij

Ener

gy (a

.u.)

1 2 3 4 5 6 7 8

-1.2E-03

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

Alija

h’s

mos

t diff

use

func

tion

H3+ [H H H]+ 2H+H+

20 c

m-1

Alijah et al.

Our work (ECGs)

Ener

gy D

iffer

ence

(a.u

.)

Vibrational Wave function Plots

Conclusions on H3+

We developed:• ECG with analytical gradients, tested on single

point calculations• Spring method to calculate PESs, tested on a 69

point PES portion of H3+

• The code is applicable to any (ne<7) molecular system

We achieved:• Most accurate variational energies to date• Most accurate (≈ 0.01cm-1) and extensive PES

(42000 grid points) of H3+

Conclusions on H3+

To be developed:• Leading relativistic corrections• Non-adiabatic corrections• Leading QED corrections

Equivalent treatment of nuclei and electrons in H3

+

The total laboratory-frame nonrelativistic Hamiltonian:

Separating out the center of mass motion

The internal Hamiltonian Molecular atom.

Explicitly Correlated Gaussian Functions for non-BO calculations of H3

+

diatomics

H3+

or

Expectation values of the ground state non-BO energies, virial coefficients (η) , and internuclear distances for some isotopologues of H3

+ . All values are calculated for an optimized 50 term explicitly correlated Gaussian basis set and are in atomic units.

Acknowledgements Coworkers:

Pawel Kozlowski Donald Kinghorn Mauricio Cafiero Sergiy Bubin Michele Pavanello Wei-Cheng Tung

Collaborators: Alexander Alijah Nikolai Zobov Irina I. Mizus Oleg Polyansky Jonathan Tennyson

Tamás Szidarovzsky Attila Császár Max Berg Annemieke Petrignani Andreas Wolf

Support: NSF