Potential energy surfaces

Fernando PiraniDipartimento di Chimica

Universita’ degli studi di Perugia

Erice 1-7 Agosto 2005

The detailed knowledge of theInteraction V(R) is still a challenge

Presented topics:• well assessed arguments• questions under investigation• future perspectives

The detailed characterization and modelling of the intermolecular interaction requires the combination of

New experiments

Extensive ab initio calculations

Development of empirical and semiempiricalmethods

Absorption spectroscopy

Differential cross section

diffraction oscillations

rainbow

scattering angle

Integralcross section

the Lambert Beer law

glory oscillations

average value ~v-2/5

collision velocity , v

Inte

gral

cro

ss s

ectio

n, Q

(v)

Inte

ract

ion

pote

ntia

l , V

intermolecular distance, R

diffraction oscillations

rainbow

diffe

rent

ial c

ross

sec

tion

scattering angle

Scattering investigations

glory

diffractions rainbow

Q(v

)v2

/5

v

average value

average value

Perugia experimentalsetup for integral crosssection measurements

Experimental Apparatus

cell

I0

N

S

I0 IB

I

IB / I0Beam transmittance (paramagnetism)

I / I0Beam attenuation (intermolecular forces)

scatteringchamber

entrance slitto the detector

ionizer

quadrupolemass filter

electronmultiplier

beam defining slit

60cm 32cm

supersonicbeam source

velocityselector

skimmers

cryostat

50cm 12cm 38cm

Stern-Gerlachmagnet

chopper

velocity selection

magnetic analysis

scattering experiments

V(R)

R

π

0

dsin ),(2

1RVV

(O2)=1.60 Å3

(Ar)=1.64 Å3

If the molecule rotate faster then thetime required for a collision, an effectiveaveraged interaction drives the collision

Ar-Kr

O2-Kr

(b) 10 K

(a) 1000 K

F.Pirani et al. JCP 75, 1042 (1981)

O2-Kr

The scattering of aligned molecules: the anisotropy in van der Waals interactions

V. Aquilanti et al., JCP, 109, 3898 (1998)

The glory shift: a signature of an embryonic H-bond

V. Aquilanti et al., Angew. Chem. Int. Ed., 117, 2408 (2005)

αH2O=1.47 Ǻ3

αO2=1.60 Ǻ3

The glory shift and quenching: the role of additional components to vdW

The glory quenching and its modification with the controlled change in the sublevels of the Cl atoms provide information on the spherical component V0 (vdW) and on the interaction anisotropy V2

V2 – configuration

interaction

Kr+Cl-

Kr Cl

Ab initio methods

Supermolecule Approach

VAB = EAB - EA - EBwhere EAB denotes the energy of the supermolecule andEA and EB the partners energies

Perturbation Theory

VAB = Vex.rep.+ Vch.tr.+ Vind+ Vdisp+ Velectr + …

Semiempirical and empirical methods

• Identification of the leading interaction components

VAB = Vex.rep.+ Vind+ Vdisp + Vch.tr.+ Velectr = VvdW + Vch.tr.+ Velectr

Semiempirical:representation of each component by theoretical formulas where some quantities are identified with basic properties of involved partners

Empirical:representation of each component by empirical formulas given in terms of fundamental physical properties of involved partners (polariz., charge, permanent multipole, ioniz. potential, electron affinity…)

The polarizability properly scales both attraction and repulsion

Further applications:

ion–neutral Chem.Phys.Lett. 183, 297 (1991)multicharged ion–neutral and ion–ion Chem.Phys. 209, 299 (1996)Atom (ion)-polyatomic molecule – Chem.Phys.Lett 350, 286 (2001); Chem.Phys.Lett. 394, 37 (2004)

When only van der Waals forces are operative!

Neutral-neutral J. Chem. Phys., 95, 1852 (1991)

attraction ~ R-6 (n-n) || ~ R-4 (i-n)

repulsion

well region

pote

nti

al en

erg

y

intermolecular distance

1/3

mole

cu

lar

volu

me

molecular polarizabilityatomic radius

εRm

Models for the representation of intermolecular forces

095.0

3/13/1

767.1attraction

repulsion

BA

BAmR

2/12/16 7.15

B

B

A

A

BA

NN

C

6672.0mR

C

Other references:ion–neutral Chem.Phys.Lett. 183, 297 (1991)multicharged ion–neutral and ion–ion Chem.Phys. 209, 299 (1996)atom-polyatomic molecule – Chem.Phys.Lett. 350, 286 (2001)Chem.Phys.Lett. 397, 37 (2004)

van der Waals forces

J. Chem. Phys., 95, 1852 (1991)

~100 systems investigated (Rm<3% <15%)

Closed shell-closed shell

NEUTRAL A – NEUTRAL B (polarizabilities αA, αB)

(Ion charge Z and polarizabilities αI, αN)

+ vdW

(repulsion + induction + dispersion)

non resonant(excimers, dications, …)

resonant at crossing(harpooning, …)

resonant at all R(H2

+,Ar2+, …)

The configuration interaction in rare gas-oxides, RgO

and in rare gas-halides, RgX

Rare gas sulfides: the interaction anisotropy

van der Waals + charge transfer

The electrostatic component is important in systems, such as:

Alkali halides

• Na+ + Cl- (charge—charge) V. Aquilanti, D. Cappelletti, F. Pirani Chem. Phys., 209, 299 (1996)

Excimers

• Kr+ + Cl- (charge—charge) M. Krauss, J. Chem. Phys., 67, 1712 (1977) (charge--quadrupole)

++

XX++((33P)P)HH++

proton-induced dipoleproton-induced dipole

proton-quadrupoleproton-quadrupoleCoulombCoulomb

XX2+2+((44S)S)HH

dispersiondispersion

inductioninductionexchange repulsionexchange repulsion

electron transfer couplingelectron transfer coupling

HX+2- Double photoionization of HX (X=Cl,Br,I)

1

Low lying states of molecular dications HX++

2 4 6

28

30

32

34

36

38

40

42

POT

EN

TIA

L E

NE

RG

Y (

eV)

INTERNUCLEAR DISTANCE (Å)

0 0 2 4 6 0 2 4 6

HCl2+

HBr2+

HI2+

3

1

1

3

1

1

3

1

1

He*(2He*(211S) + NS) + N22OO

He*He*…… NNO NNO

NNO NNO …… He* He*

The atom (ion) — molecule case

As usual VAB = Vex.rep.+ Vind + Vdisp + Vch.tr + Velectr = VvdW + Vch.tr. + Velectr

Prototypical examples are: F, Cl – H2 V. Aquilanti et al, JPC A, 105, 2401 (2001) Ar+ - N2 R. Candori et al, JCP, 115, 8888 (2001)

Vch.tr. depends on the overlap between orbitals which exchange the electron

(exponentially decreasing with R and varying with the relative orientation of orbitals involved in the exchange)

Velectr relates to the charge distribution on the molecular frame (obtainable from

ab initio calculations)

VvdW arises from the combination of size repulsion effects (short range) with

dispersion and induction attraction (long range). It is very difficult to assess and to model such component

22--

++

++

1b1

(2p non bonding orbital)

3a1

(sp2 lone pair orbital)

Atom: VAtom: Vvdwvdw + V + Vch.tr.ch.tr.

Ion: VIon: Vvdwvdw + V + Velectrelectr + V + Vchtrchtr

Atom (ion) - water

Atom (ion) - benzene

The atom (ion) — molecule case

The assessment of the strength of the van der Waals component involves again the characterization of its dependence on the molecular polarizability (related to features of the electronic distribution in the HOMO and LUMO orbitals). For small and homonuclear diatoms (H2 , N2 …) the electronic distribution is approximately representable through an ellipsoid whose dimensions depend on the tensor components of the polarizability (a single dispersion-induction center).

For big and homonuclear diatoms (I2 , Br2 …) the electronic distribution is better represented by a combination of ellipsoids defined in terms of molecular polarizability contributions: one associated to the bond and the other to the lone pairs(multiple dispersion-induction centers).

Heteronulear diatoms (HCl, HBr …) fall in the two previous cases.

A polyatomic molecule can be considered as the combination of bond and lone pair components (multiple dispersion-induction centers). For the separability of molecular polarizability into several tensor components see JCP, 32, 502 (1960)

The atom (ion) — molecule case

The proper modelling of the van der Waals component requires the development of atom (ion)—bond and atom (ion)—lone pair potential models exhibiting two basic features:

• they must involve an unique potential function, defined in terms of few parameters, each one having a specific meaning;

• they must remove, totally or partially, the inadequacies, both at short and at long range, of the “venerable” LJ models

Atom(ion)-bond formulation of the potential energy function (1)

V (r,) ()m

n(r,) m

rm ()

r

n(r,)

n(r,)

n(r,) m

rm ()

r

m

n(r,) 4.0r

rm ()

2

rm () rmsin2 rm//

cos2

() sin2 // cos2

A B

C

r

m= 6 atomm= 4 ion

CPL, 394, 37 (2004)

V VAA

VA C

ii1

6 V

A Hii1

6

Two-body interactions

Three body interactions

V VAA

VABC

VA C

ii1

6 V

A Hii1

6 V

A CiCi1i1

6 V

A CiHi6i1

6

Potential energy function: Atom-bz

Atom(ion)-bond formulation of the potential energy function (2)

VAtom benzene V

A CC

ii1

6 V

A CH

jj1

6

Vion benzene V

ion CC

ii1

6 V

ion CH

jj1

6 V

ion quadrupole

Vi Vel 1

4o

qiqionri ion

Vion quadrupole Vi

i1

18

Benzene-Ar system

Benzene-Cl- and Benzene-K+clusters

6.0

4.5

3.0

1.5

0.0

y /

Å

-4.0 -2.0 0.0 2.0 4.0x / Å

-300 meV

6.0

4.5

3.0

1.5

z /

Å

-300 meV

6.0

4.5

3.0

1.5

0.0

y /

Å

4.02.00.0-2.0-4.0x / Å

-114 meV

6.0

4.5

3.0

1.5

z /

Å

-830 meV

Atom-bond formulation vs ab initio calculations:

the case of K+ – C6H6

(Benzene)m-ion-Atomn clusters

V VI (benzene)

ii1

m V

Aj (benzene)

ij 1

n

i1

m V

Aj I

j 1

n V

Aj Akk j

n

j 1

n 1

Vion benzene V

ion CC

ii1

6 V

ion CH

jj1

6 V

ion quadrupole

VA benzene V

A CC

ii1

6 V

A CH

jj1

6

Ar-Bz-Cl- and Ar-Bz-K+clusters: Potential energy surfaces

6.0

4.0

2.0

0.0

-2.0

-4.0

z/ Å

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0x/ Å

-1060 meV

-980 meV

K+

Only one isomer

Two isomers

4.0

2.0

0.0

-2.0

-4.0

z/ Å

8.06.04.02.00.0-2.0-4.0x/ Å

Cl -

-400 meV

Ar-Bz-Cl- and Ar-Bz-K+clusters

Ar-Bz-Cl-

Ar-Bz-K+

The PES’s and related force fields have been extensively exploited in molecular dynamics simulations

DL-POLY programs

Cluster configuration energy (function of Etotal) defined, using the present method, as a sum of fragment contributions

opening of isomerization channels beginning of dissociation

Conclusions and perspectives

Present investigations open new questions and further perspectives:

- Molecule considered as non-rigid body

crucial is the dependence of VAB on the internal coordinates ri

the basic point is the characterization of the dependence of the polarizability

on ri

- Molecule-Molecule systems

Empirical study of the polarizability α and its anisotropy Δα/α for diatomic bonds and molecules

α = non bonding electron + bonding electron effective bond excitation contributions contributions order function

The above terms depend on: - the number of non bonding and total valence electrons of Atom 1 || || Atom 2

-polarizability value of Atom 1 and Atom 2

- distance r and equilibrium distance re

The method provides correct values for several homonuclear and heteronuclear bonds

Δα/α defined in a similar way

N2 – N interaction potential from polarizabilities

r

R

atom-bond model

atom-atom model

Diatom-diatom interaction potentials from polarizabilities

Dynamical simulations

•The benzene molecule is considered as a rigid body.

• A time step of 1 fs has been adopted to integrate the equa-tions of the motion.

•Dynamical simulations have been performed in the context of the microcanonical ensemble.

v(t+t/2) v(t-t/2) + t F(t)/mr(t+t) r(t) + t v(t+t/2)

v(t)=0.5*(v(t+t/2) + v(t-t/2))

Ti mivi

2

i1

N

kB f

Energy contributions vs Etotal

-920.0

-910.0

-900.0

-890.0

-880.0

EK

+-b

z / m

eV

300.0250.0200.0150.0100.050.00.0T/ K

-110.0

-105.0

-100.0

-95.0

-90.0

EK

+-A

r

/ m

eV

-30.00

-25.00

-20.00

-15.00

Ebz

-Ar

/ m

eV

-1040.0 -1000.0 -960.0 -920.0Etotal / meV

Ebz--Ar

EAr--K+

Ebz—K+

Isomerization (1)

0.25

0.20

0.15

0.10

0.05

tim

e (1

|1)

/ ti

me(

2|0)

275.0270.0265.0260.0255.0250.0245.0240.0

T / K

Isomerization (2)

160

140

120

100

80

60

40

20

350300250200150100500

150

150

120

120

120 120

120 90

60 60 60

30 30

90

O

CH3

CH3

Ar

Rcm

cm

0 -267 3.59 70 90

61 -206 3.95 23 0 / 180

69 -198 3.92 0 All

91 -176 4.18 180 All

E /cm-1 Eabs /cm-1 Rcm /Å /deg /deg

112 -155 4.86 120 0 / 180

Isoenergetic countours and main stationary properties of the atom-bond PES of DME-Ar

The points of the 2D cross section have been taken at the minimum along Rcm

150

100

50

0

350300250200150100500

-19

-19

-19

-19

-20 -20

-21

-21

-21

-21

-23

-23

-23

-

23 -23

-24

-24

-24

-24

-25

-25

-25

-2

6

-26

-27

-27

Isoenergetic countours of the atom-bond PES of H2O-He

The points of the 2D cross section have been taken at the minimum along Rcm

Comparison between experimental and atom bond spherical average parameters of H2O-He

-23.243.083.44At-bond

-22.193.063.45Exper.

ε / cm-1 / ÅRm / ÅModel

Comparison between experimental and atom-bond integral and differential cross section of H2O-He