Quantum Monte Carlo calculations on open shell species inside helium clusters

(1)

HAL Id: hal-00832967

https://hal-upec-upem.archives-ouvertes.fr/hal-00832967

Submitted on 11 Jun 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quantum Monte Carlo calculations on open shell species inside helium clusters

Marius Lewerenz

To cite this version:

Marius Lewerenz. Quantum Monte Carlo calculations on open shell species inside helium clusters.

2013. �hal-00832967�

(2)

Quantum Monte Carlo calculations on open shell species inside helium clusters

Marius Lewerenz

Laboratoire de Modélisation et Simulation Multi Echelle UMR8208 CNRS

Université Paris Est (Marne la Vallée) 5, Blvd. Descartes, Champs sur Marne

77454 Marne la Vallée Cedex 2 France

Marius Lewerenz Helsinki, 13.11.2012 2

Acknowledgments

Paris-Est:

Ji Jiang, Ph.D student, Ar

⁺

@He

_n

, I

^q

@He

_n

Mirjana Mladenović, CO

⁺

@He

_n

Prague:

Petr Slavíček, Pb

^q+

@He

_n

ANR project DYNHELIUM (Toulouse, Rennes, Paris)

(3)

•Helium-helium interaction is of weak van der Waals type, closed shell atoms of very low polarisability, D

_e≈7.6 cm^-1

•Helium atoms have a relatively small mass.

•Large zero point energy effects (D

₀

for He

₂≈ 0.001 cm^-1

).

•Helium clusters are small chunks of a quantum liquid.

•Quantum statistical effects: bosonic

⁴

He, fermionic

³

He.

•Superfluidity

in bulk liquid

⁴

He below 2.17 K, in

³

He at mK level

•A very special solvent: Is there a new chemistry?

•Implantation of dopants through (multiple) inelastic collisions.

•Weak interactions with dopant.

•Binding energy and position of dopants depend on quantum effects.

Delicate balance between potential and quantum kinetic energy

What makes helium clusters interesting?

Plenty of interesting experiments but theoretical difficulties!

•Matrix spectroscopy with minimal perturbations:

OCS, (HF)

_n

, biomolecules at 0.4 K, radicals

•Reaction dynamics at very low temperatures: Ba + N

₂

O → BaO + N

₂

•Preparation of reactive intermediates: HF ··· CH

₃

, HCN

··· CH₃etc.

•Preparation of high spin metal polymers: Na

₃

, K

₃

, Rb

₃

etc.

•Assembly of cold clusters: Ag

_n

, Mg

_n

•Thermodynamically unstable isomers: linear (HCN)

_n

•Nanomodels for molecule-surface interactions: HCN···Mg

₃

etc.

•Container for soft ionisation for analytical mass spectrometry?

•Energy dissipation by coupling to the bath?

•Confinement medium for cluster ignition and Coulomb explosion.

•Spacer for interatomic Coulombic decay (ICD).

Recent applications of helium clusters

(4)

A typical helium droplet experiment

He_n D@He_n (partial) destruction of cluster

Observation of ionic clusters resulting from fragmentation or ejected photo fragments

Marius Lewerenz PRAHA2012, 8.9.2012 6

I@He _n from CH ₃ I → CH ₃ + I.

Experimental fragment distributions vs.

ab initio+diffusion quantum Monte Carlo results

I@He_n, with SO coupling Incremental binding energies

Largely isotropic, no clear shells

Braun and Drabbels 2007 n I@He_n

(5)

Pair potentials involving helium and metals

He-He He-Ag

He-Mg He-Na

Shallower well than He-Heand larger equilibrium distance for He-M

Ions in helium clusters

• Massive change of interaction potential

• Polarisation forces

• Enhanced localisation of helium atoms

• “Snowball’’ formation

Two step photoionisation Theisen et al., TU Graz

(6)

Modelling open shell species inside helium clusters

ANR project DYNHELIUM (Paris , Rennes, Toulouse):

Dissociation, recombination/caging and/or cluster exit of reaction products Our ultimate test problem, well known in the gas phase:

Photodissociation of CH

₃

I → CH

₃

+ I inside He

_n

We need global potential energy surfaces for ground and excited CH₃I@He_nand for the relevant fragments CH₃@He_nand I@He_n (electronic anisotropy!).

Isoelectronic warm up system:

Ar⁺ions (s²p⁵valence shell, X²Σ⁺and A²Πstates for Ar⁺He).

Spin-orbit coupling between Σ and Πstates has to be included in the model.

Spin-orbit splitting is typically larger than the van der Waals interaction:

∆= 1432 cm^-1for Ar⁺, ∆= 7600 cm^-1for I

Include non additive induced dipole – induced dipole interaction for charged species

Diffusion quantum Monte Carlo (DMC)

•Isomorphism between time dependent

Schrödinger equation and a multi dimensional diffusion equation (Fermi, Ulam)

•Exact solution except for statistical errors

Solution by propagation of an ensemble of random walkers in imaginary time Cartesian coordinates, precision σ

_E

/E = 10

^-6

– 10

^-3

?

(7)

DMC calculations for Ar ⁺ He _n

Potential model:

Anisotropy due to Ar

⁺

s

²

p

⁵

valence shell → X

²Σ⁺

and A

²Π

states for Ar

⁺

He.

IP(Ar)=15.76 eV

→ He⁺

+Ar channel is unimportant, single configuration.

RCCSD(T) calculations with (aug)-cc-pVXZ basis sets (MOLPRO).

Infinite basis set ab initio points fitted to HFD-style analytical form with fixed C

₄

coefficient computed from α

_He

= 1.41 a

₀³

.

Strong spin-orbit interaction in Ar

⁺

(∆ = 1432 cm

^-1

):

Non additive many body potential model including induced dipoles on He with additional spin-orbit mixing included using atomic

∆_Ar+

(complex 6 x 6 matrix to diagonalise in each DMC step).

_.

Ar ⁺ He: convergence of interaction energy

RCCSD(T) calculation, standard and augmented basis sets

RCCSD(T)/VXZ RCCSD(T)/aVXZ

(8)

Ar ⁺ He: spectroscopic observables

CCSD(T), infinite basis extrapolated potentials (aQ56), atomic ∆

^SO

, variational rovibrational calculation in Laguerre basis,

⁴

He

⁴⁰

Ar

⁺

Expectation values for rotational constants in cm-1 Vibrational transition frequencies in cm-1

Our Ar

⁺

He potential is excellent !

This work This work This work

Ar ⁺ He _n : DMC ground state energies vs. exp.

Total energies extrapolated to ∆τ=0 Energy increments

Spin orbit coupling is responsible for magic character of n=12 cluster Magic number

at n=12

Fragmentation after ionisation of Ar@He_N, Brindle et al. 2005

Our DMC calculation

(9)

DMC calculations for I@He _n

Motivation: Photodissociation of CH

₃

I → CH

₃

+ I inside He

_n

.

We need global potential energy surfaces for ground and excited CH

₃

I@He

_n

and for the relevant fragments CH

₃

@He

_n

and I@He

_n

Potential model:

Anisotropy due to I s

²

p

⁵

valence shell → X

²Σ⁺

and

A²Π

states for I-He.

RCCSD(T) calculations with aug-cc-pVXZ basis sets and relativistic pseudopotential (ECP) from K. Petersen.

Ab initio points fitted to extended Tang-Toennies analytical form.

Very strong spin-orbit interaction in I:

Non additive many body potential model with additional spin-orbit mixing using atomic ∆

_I

(complex 6 x 6 matrix to diagonalise in each DMC step).

∆_SO

dominates so much over E

_vdW

that SO mixing is almost perfect!

r/a₀ r/a₀

Eint/cm-1

I-He: Convergence of RCCSD(T)/ECP calculations

I-He X

²

Σ

⁺

I-He A

²

Π

Spin-orbit coupling mixes the ²Σ_½and ²Π_½components: 6x6 complex matrix

(10)

I-He: Interaction potential with SO coupling

X²Σ⁺ A²Π

X A₁

A₂

Strong Σ/Πmixing makes I-He interaction

almost isotropic Eint/cm-1

r/a₀

RCCSD(T), small core ECP, MOLPRO code, infinite basis extrapolation, experimental atomic SO constant.

I@He _n : Radial helium density from DMC

ρHe/Ǻ-3

r/Ǻ

2^nd“shell” starting near n=20

ρ_LHe

(11)

I@He _n : Incremental binding energies from DMC

I@He_n, with SO coupling

Largely isotropic, no clear shells

n Braun and Drabbels 2007

I@He_n

I ^- @He _n : Binding energy and radial helium density

ρHe/Ǻ-3

(En-En-1)/cm-1

Soft transition to 2^nd“shell” near n=25

(12)

I ^- @He _n : Angular correlations

Probability density for He-I-He angle α

Squeezing enhances structure but does not induce localization

α/radian P(α)

I^-@He₁₀ I^-@He₂₀

I ²⁺ @He _n : Binding energy and radial helium density

17

17 30 1

6 14

12 ^ρ

He/Ǻ-3

r/Ǻ (En-En-1)/cm-1

n

2^nd“shell” starting at n=17

(13)

Evidence for formation of CO⁺Heions in several drift tube experiments.

No experimental spectroscopic information.

Mixed cluster ions of the composition CO⁺He_nshould be accessible in drift tube experiments, mixed gas expansions coupled to electric discharges, or CO

ionization inside large He clusters.

Ionisation of CObarely changes the rotational constants but strongly affects the interaction with helium: CO@He_nand CO⁺He_nare an ideal pair to understand rotation in helium clustersby separating effects due to mass and interaction.

Potential surface can be checked by ion depletion spectroscopy (see N₂⁺-He_n)^.

Astrophysical motivation

COis rather abundant in interstellar space andCO⁺has been identified in 1993.

Low energy collisions with helium atoms, its second most abundant and non reactive collision partner, are governed by the weak intermolecular interaction leading to the van der Waals complexHe-CO⁺.

CO ⁺ -He and CO ⁺ @He _n

2D contour plot of the RCCSD(T) PES.

Contour lines at intervals of 25 cm^-1, first contour placed at -250 cm^-1. Theblue lineshows the variation of the Jacobi distance R along the minimum energy

θθθθ.

Features of the CO ⁺ -He surface

RCCSD(T) surface extrapolated to complete basis set limit.

V

^min

= -275.3 cm

^-1

E

⁰

= -195.0 cm

^-1

A

⁰

= 7.168 cm

^-1

B

⁰

= 0.466 cm

^-1

C

⁰

= 0.411 cm

^-1

Quasilinear molecule with strong permanent dipole moment and strong IR transition moment.

Low energy scattering resonance.

(14)

Spectroscopic results from DVR-DGB calculations

He-CO⁺(X²Σ⁺) 2D RCCSD(T) potential energy surfaces V(R,θ) at r(CO)=1.11783 Å

7.7 cm^-1

35 cm^-1 73 cm^-1 106 cm^-1 134 cm^-1

0 cm^-1

93 cm^-1 124 cm^-1 145 cm^-1 152 cm^-1

160 cm^-1

n^0,0=0 n^0,0=1 n^0,0=2 n^0,0=4 n^0,0=6

n^0,0=8 n^0,0=7

n^0,0=5 n^0,0=3

n^1,1=1

n^0,0=9

Contour plots of 2D (ro)vibrational wavefunctions obtained in 2D DVR- DGB calculations. Contours are drawn at intervals of 5% of the maximum wavefunction amplitude.

(15)

CO ⁺ -He ground state as seen by DMC:

He density histogram in cylinder coordinates (z,r)

O→

→→→C defines z-axis, origin at c.o.m. of CO unit

Contour lines at ρ_He= 0.001, 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.15, 0.20, 0.25 Ǻ^-3 ρ_He/Ǻ^-3

O C

^r/Ǻ

z/Ǻ

E0= -194.95 ±±±±0.05 cm^-1

Blue shift/cm-1∆∆∆∆E/cm-1

Energy increments ∆∆∆∆E (chemical potential, bottom graph) and CO⁺ frequency shift (upper graph) from adiabatic method (intermolecular potential as parametric function of the CO⁺ vibrational state).

Many body model with induced dipoles.

Note the turn-around of the frequency shift at the last “magic” size.

CO ⁺ @He _n

(16)

CO ⁺ He _n ground state densities from DMC

n=4 n=5

n=6 n=7

r/Ǻ

r/Ǻ r/Ǻ

z/Ǻ r/Ǻ z/Ǻ

z/Ǻ z/Ǻ

ρ_He/Ǻ^-3

n=10 n=11

n=12 n=15

O C O C

CO ⁺ He _n DMC ground state densities:

build up of strongly anisotropic first helium shell

(17)

CO ⁺ He _n DMC ground state densities:

onset of second helium shell at n=16

n=16

O C

^r/Ǻ

z/Ǻ ρ_He/Ǻ^-3

n=30

n=40 n=50 n=25

r/Ǻ ρ_He/Ǻ^-3

CO ⁺ He _n DMC ground state densities:

build up of second helium shell

O C O C

(18)

CH ₃ -He

Ab initio RCCSD(T)calculations with aug-cc-pVXZbasis sets (X=D,T,Q,5) CH₃keeping C_3vsymmetry and fixed C-H distance: only umbrella angle α Heposition relative to CH₃center of mass in spherical coordinates R,θ,φ Several 3D surfaces assembled into 4D surface including CH₃relaxation Overall about 3000 potential energy points

Analytical representation with angle dependent HFDform expanded over real spherical harmonics T_lmwith symmetry restrictions on l,m:

V(R,θ,φ) = A exp{-b(θ,φ) [R-R_e(θ,φ)]} –Σ_kC_k(θ,φ)/R^k X(θ,φ) = Σ_lmx_lmT_lm(θ,φ) X=b, R_e, C_k

500-1000 points per 3D cut are fitted with 38 parameters and rms < 0.1 cm^-1

•Effective rotational constants for CO

⁺

in helium (DMC/PIMC in collaboration with P. N. Roy).

•CH

₃

radicals in helium, reactive complexes.

•Photodissociation of CH

₃

I and CF

₃

I (ZPAD, DMC etc.)

•Dopant spectroscopy (Mg

^*

, Ag

^*

, Ag

⁺

etc.).

•Transport properties (Mg

⁺

, Na

⁺

).

•DMC and ZPAD calculations on Xe

_n

He

_m

.

•DMC with constraints ((H

₂

)

_n

, He

_n

(H

₂

)

_m

Quantum Monte Carlo calculations on open shell species inside helium clusters

HAL Id: hal-00832967

https://hal-upec-upem.archives-ouvertes.fr/hal-00832967

Quantum Monte Carlo calculations on open shell species inside helium clusters

Marius Lewerenz

To cite this version:

Quantum Monte Carlo calculations on open shell species inside helium clusters

Marius Lewerenz

Laboratoire de Modélisation et Simulation Multi Echelle UMR8208 CNRS

Université Paris Est (Marne la Vallée) 5, Blvd. Descartes, Champs sur Marne

77454 Marne la Vallée Cedex 2 France

Acknowledgments

Paris-Est:

Ji Jiang, Ph.D student, Ar

@He

, I

@He

Mirjana Mladenović, CO

@He

Prague:

Petr Slavíček, Pb

@He

ANR project DYNHELIUM (Toulouse, Rennes, Paris)

•Helium-helium interaction is of weak van der Waals type, closed shell atoms of very low polarisability, D

•Helium atoms have a relatively small mass.

•Large zero point energy effects (D

for He

).

•Helium clusters are small chunks of a quantum liquid.

•Quantum statistical effects: bosonic

He, fermionic

He.

in bulk liquid

He below 2.17 K, in

He at mK level

•Implantation of dopants through (multiple) inelastic collisions.

•Weak interactions with dopant.

•Binding energy and position of dopants depend on quantum effects.

Delicate balance between potential and quantum kinetic energy

What makes helium clusters interesting?

Plenty of interesting experiments but theoretical difficulties!

•Matrix spectroscopy with minimal perturbations:

OCS, (HF)

, biomolecules at 0.4 K, radicals

•Reaction dynamics at very low temperatures: Ba + N

O → BaO + N

•Preparation of reactive intermediates: HF ··· CH

, HCN

•Preparation of high spin metal polymers: Na

, K

, Rb

etc.

•Assembly of cold clusters: Ag

, Mg

•Thermodynamically unstable isomers: linear (HCN)

•Nanomodels for molecule-surface interactions: HCN···Mg

etc.

•Container for soft ionisation for analytical mass spectrometry?

•Energy dissipation by coupling to the bath?

•Confinement medium for cluster ignition and Coulomb explosion.

•Spacer for interatomic Coulombic decay (ICD).

Recent applications of helium clusters

A typical helium droplet experiment

I@He n from CH 3 I → CH 3 + I.

Experimental fragment distributions vs.

ab initio+diffusion quantum Monte Carlo results

Pair potentials involving helium and metals

Ions in helium clusters

Modelling open shell species inside helium clusters

ANR project DYNHELIUM (Paris , Rennes, Toulouse):

Photodissociation of CH

I → CH

+ I inside He

Diffusion quantum Monte Carlo (DMC)

•Isomorphism between time dependent

Schrödinger equation and a multi dimensional diffusion equation (Fermi, Ulam)

•Exact solution except for statistical errors

Solution by propagation of an ensemble of random walkers in imaginary time Cartesian coordinates, precision σ

/E = 10

– 10

I@He _n from CH ₃ I → CH ₃ + I.

DMC calculations for Ar ⁺ He _n

Ar ⁺ He: convergence of interaction energy

Ar ⁺ He: spectroscopic observables

Ar ⁺ He _n : DMC ground state energies vs. exp.

DMC calculations for I@He _n

I@He _n : Radial helium density from DMC

I@He _n : Incremental binding energies from DMC

I ^- @He _n : Binding energy and radial helium density

I ^- @He _n : Angular correlations

I ²⁺ @He _n : Binding energy and radial helium density

CO ⁺ -He and CO ⁺ @He _n

Features of the CO ⁺ -He surface