Non-adiabatic calculations for rovibrational states of H2...

logo

Non-adiabatic calculations for rovibrational states ofH+

2 , H2 and H+3

orNon-adiabatic theory in terms of a single potential energy

surfaceor

Which masses are moving in the vibration or rotation of amolecule?

Ralph Jaquet,Theoretical Chemistry, University Siegen, Germany

February 10, 2012

logo

Introduction Distant-dependent effective masses Coupled surfaces Conclusion

Outline

1 Introduction

2 Distant-dependent effective masses

3 Coupled surfaces

4 Conclusion

logo


Motivation: Reaction dynamics and spectra of Hydrogen ionclusters (with all isotopologues)

H+3 : H+ + H2 and H + H+

2 (and elect. excited states)

Reaction on the ground state PES: strong influence of non-adiabatic effectsfor Ecoll > 1eV: avoided crossing and conical intersections

Spectrum for the ground state PES (rovibrational transitions): deviations toexperiment for E(BO) or E(BO)+ E(ad): much too large → ’weak’non-adiabatic effects

logo


PES: global and local regions, adiabatic and non-adiabaticcorrections

Comparison ’exp-calc’ with Pavanello et al, PRL, 2012 (PA12)

exp BO BO DAR+NA DAR+NA DAR+NAPA12 PA12 pw PA12 pw pw

VRM2 VRM1 VRM2GG/CIR12 GG/GG GG/GG/CIR12

13638.251 3.54 3.45/ 3.51 -0.22 -0.44/-0.49 -0.90/-0.95/-0.2915058.680 2.14 2.01/ 2.09 0.15 -0.13/-0.01 -0.60/-0.58/0.0215130.480 1.03 0.59/ 0.60 0.03 0.34/ 0.53 -0.10/-0.0415450.112 -1.48 -1.73/-1.55 -0.06 0.30/ 0.54 -0.23/-0.0915643.052 -0.29 -0.41/-0.37 0.04 0.56/ 0.64 -0.07/0.0115716.813 1.08 0.87/ 1.01 0.16 -0.24/ 0.46 -0.14/-0.1616506.139 0.49 0.29/ 0.43 0.23 0.75/ 0.74 0.05/0.0416660.240 -0.50 -0.71/-0.58 0.02 0.44/ 0.47 -0.20/-0.17

VRM1: NU(rotation), NU23(vibration)VRM2: NU(rotation), NU + 0.4753 me(vibration) (as in PA12)

pw: JCP (submitted Sept. 2011, revised 2012)

logo


PES: global and local regions, adiabatic and non-adiabaticcorrections

H3+: lowest three PESs

01

23

45

67

8910

r23(H - H)

0 1 2 3 4 5 6 7 8 9 10

r21(H - H)

0

20000

40000

60000

80000

100000

120000

140000

H3+: strong non-adiabatic coupling

ground - 1st exc. state1st - 2nd exc. state

01

23

45

67

89

10

R210 1 2 3 4 5 6 7 8 9 10R23

00.10.20.30.40.50.60.70.80.9

1

Three lowest PESs (H+3 ): Avoided crossings and conical intersections→

non-adiabatic effects

logo


Molecular Schrodinger equation: BO → NAD

HΨ(~R,~r) = EΨ(~R,~r) (1)

H = Tn + Te + Ven + Vee + Vnn = Tn + Hcn (2)

Non-adiabatic ansatz:

Ψ(~R,~r) =∑p,q

cp,qχp(~R)ψq~R(~r) (3)

How to take care of non-adiabatic effects?

weak effects: stay with one PES and take distant-dependent effectivemasses

strong effects: coupling of several PESs

logo


NAD(1): weak non-adiabaticity

Non-adiabatic ansatz:(1) exact separation in all possible dissociation limits,(2) which reduces to the adiabatic approximation in the limit M →∞

Ψg = ψgχ(~R) +1

M + 1~ω(~r) · ∇χ (4)

~ω(~r): vector of excited electronic wave functions∇χ: vector of excited vibrational functions

If we define~ω(~r) = Ng[ψ(~ra)~ra − ψ(~rb)~rb] (51)

we can write

Ψg = ψgχ(~R) +1

M + 1~ω(~r) · ∇χ (52)

This is obviously in the form of a non-adiabatic ansatz, with ~ω(~r) a vector of(unnormalized) excited electronic wave functions and ∇χ a vector of excitedvibrational functions.

.

ω

ω

ψ z

xx

z

g

25

Schematic representation of the function ψg , ωz , ωx (Kutzelnigg, R.J., 2007,2008).

logo


NAD(2): weak non-adiabaticity

H = H0 + Hp, Hp = − ~2

MR∇e

R · ∇nR , (5)

Perturbation Hp defines the so-called mixed derivative.Using the cartesian coordinates for the relative nuclear motion ~R(X ,Y ,Z ):∇R = ( ∂

∂X~nX + ∂

∂Y~nY + ∂

∂Z~nZ ) k : nuclear states, λ: electronic states

Herman, Asgharian (1966):

∆E (2)λk = − ~4

M2R

⟨k

∣∣∣∣∣∣∑λ′ 6=λ

∑i=X ,Y ,Z

〈λ|∇ei |λ′〉∇n

i 〈λ′|∇ei |λ〉∇n

i

(Eλ′ − Eλ)

∣∣∣∣∣∣ k⟩. (6)

A(R) (for vibration: Z ), B(R) (for rotation: X , Y )

A(R)

M=

2~2

MR

∑λ′ 6=λ

〈λ|∇Z |λ′〉〈λ′|∇Z |λ〉(Eλ′ − Eλ)

,B(R)

M=

2~2

MR

∑λ′ 6=λ

〈λ|∇X |λ′〉〈λ′|∇X |λ〉(Eλ′ − Eλ)

,

(7)

logo


The ’vib-mass’ in the MO-LCAO approx. and PT

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

A

R(H2+)

H2+: vib. mass factor

A (LCAO alpha=1)

A (LCAO alpha=FH)

A (H2+ in H3+)

logo


The ’rot-mass’ in the MO-LCAO approx. and PT

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 2 4 6 8 10

B

R(H2+)

H2+: rot. mass factor

B (LCAO alpha=FH)

B (LCAO alpha=semi-emp)

B (LCAO FH(pol))

B (H2+ in H3+)

logo


The ’vib-rot-mass’ in H2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 2 4 6 8 10

A/B

R(H2)

B

A

H2: vib. and rot. mass factors

A (LCAO alpha=HL)

A (LCAO alpha=ROSEN)

A (LCAO alpha=ROSEN(MOD))

A (PK-2009)

B (LCAO alpha=HL)

B (LCAO alpha=ROSEN)

B (LCAO alpha=ROSEN(MOD))

B (PK-2009)

logo


H+2 , J=0 and 20: Non-adiabatic corrections

[ cm−1] Moss M1 Mnew (H+2 in H+

3 ) Moss M1 MnewJ=0 ∆ ∆ J=20 ∆ ∆

v=0 0.109 -0.012 -0.002 | 0.555 0.337 -0.0211 0.311 -0.028 -0.004 | 0.713 0.326 -0.0182 0.492 -0.033 -0.002 | 0.854 0.315 -0.0153 0.657 -0.033 0.000 | 0.977 0.302 -0.0144 0.806 -0.031 0.001 | 1.081 0.285 -0.0115 0.938 -0.026 0.002 | 1.162 0.266 -0.0106 1.051 -0.020 0.006 | 1.217 0.242 -0.0077 1.148 -0.016 0.007 | 1.242 0.214 -0.0058 1.225 -0.013 0.009 | 1.230 0.183 -0.0049 1.280 -0.012 0.010 | 1.171 0.148 -0.003

10 1.311 -0.013 0.011 | 1.049 0.111 -0.00211 1.315 -0.017 0.011 | 0.826 0.071 -0.00212 1.287 -0.022 0.01113 1.222 -0.027 0.01014 1.114 -0.030 0.01015 0.956 -0.030 0.00816 0.742 -0.027 0.00617 0.466 -0.018 0.00418 0.154 -0.006 0.00119 0.009 0.010 0.011

logo


H2, Non-adiabatic corrections (∆G(v + 1/2) in cm−1)

J=0 PT ∆ 10 PT ∆ 20 PT ∆ 30 PT ∆v=0 0.8365 -0.0050 0 0.7853 -0.0029 0 0.6759 0.0010 0 0.5264 0.0072

1 0.7567 -0.0052 1 0.7108 -0.0025 1 0.6074 0.00142 0.6817 -0.0041 2 0.6395 -0.0020 2 0.5317 0.00303 0.6099 -0.0033 3 0.5686 -0.0013 3 0.4393 0.00444 0.5389 -0.0025 4 0.4939 0.0003 4 0.3136 0.00425 0.4642 -0.0007 5 0.4096 0.0016 5 0.1254 0.00306 0.3811 0.0006 6 0.3061 0.0022 6 -0.1862 -0.00077 0.2809 0.0014 7 0.1701 0.00198 0.1526 0.0013 8 -0.0196 0.00079 -0.0204 0.0005 9 -0.2930 -0.0026

10 -0.2594 -0.0022 10 -0.7005 -0.004911 -0.5909 -0.0046 11 -1.3811 -0.001912 -1.0436 -0.003713 -1.6374 0.0037

PT: Pachucki, Komasa (2009), ∆=E(pw)-E(PK), pw: with Mnew (H+2 in H+

3 )

logo


Effective kinetic energy operator for a triatomic molecule

H2 + H+ versus H+2 + H

second derivative terms: µ→ µeff (r ,R, θ)

Tvib = δk,k′δj,j′

[− ~2

2µvibr

∂2

∂r 2 −~2

2µvibR

∂2

∂R2 + j(j + 1)

(~2

2µvibr r 2

+~2

2µvibR R2

)]− δk,k′k2〈j ′k | sin−2 θ|jk〉

(~2

2µvibr r 2

+~2

2µvibR R2

),

Tvib,rot = δk,k′δj,j′~2 J(J + 1)− 2k2

2µrotR R2

− δk′k±1δj,j′~2

2µrotR R2 C±Jk C±jk + δk,k′k2〈j ′k | sin−2 θ|jk〉

(~2

2µrotr r 2 +

~2

2µrotR R2

)(Jacobi coordinates)

logo


The ’vib-rot-mass’ of H2/H+2 in H+

3 asymptotically

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 2 4 6 8 10

A/B

r(H2 / H2+)

H3+(R=100): vib. and rot. mass factors for H2 and H2

+

H2+

H2 A

B

B

A

A (ground-state, VIB)

A (1st excited state, VIB)

B (ground-state, ROT)

B (1st excited state, ROT)

A-PK-2009

B-PK-2009

logo


The ’vib-mass’ µ(r) (diatomic) in H+3

0

1

2

3

4

5

6

0 0.5

1 1.5

2 2.5

3 3.5

4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

me

H3+: electron mass contribution (diatomic), theta=90

Rr

me

X

0.3

0.5

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Contribution of the electron mass ∆me to the proton mass: µvibr (H2/H+

2 ) in H+3

Near the energy minimum (X=r = 1.65,R = 1.43 Bohr): ∆µvibr =0.3607 me.

logo


Coupled surfaces: Potential versus Kinetic energy coupling

a) Adiabatic calculation (Kinetic energy coupling)

adiabatic and non-adiabatic coupling matrix elements become singular(r (H2) ≈ 2.5 Bohr, R= large)

b) Diabatic calculation (Potential energy coupling)

1: (1) DIM-type potential (Kamisaka, Nakamura et al, 2002), (b)→ improvedPESs

2: Rovibrational calculations for potential in diabatic representation

3: Filter diagonalisation method (Mandelshtam and Taylor, 1995)

4: Bound states and Resonances

5: Are non-adiabatic corrections for individual rovibrational states similar inthe range 0-16000 cm−1?

logo


H+3 : Coupled PES: J=0, A2-states

-6

-5

-4

-3

-2

-1

0

1

2

6000 8000 10000 12000 14000 16000 18000 20000 22000

fre

qu

ency

dif

fere

nce

[cm

-1]

transition frequency [cm-1

]

A2: BO-ADIA A2: BO-DIAB

logo


H+3 : Coupled PES: J=0, E-states

-7

-6

-5

-4

-3

-2

-1

0

1

2

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

fre

qu

ency

dif

fere

nce

[cm

-1]


]

E: BO-ADIA E: BO-DIAB

logo


H+3 : Coupled PES: J=1, A′2- and A′′2-states (A′2: 86.966 cm−1)

-5

-4

-3

-2

-1

0

1

2

0 2000 4000 6000 8000 10000 12000 14000 16000

fre

qu

ency

dif

fere

nce

[cm

-1]

J=1: A2’: BO-ADIAJ=1: A2’: BO-DIAB

-7

-6

-5

-4

-3

-2

-1

0

1

2

2000 4000 6000 8000 10000 12000 14000 16000

fre

qu

ency

dif

fere

nce

[cm

-1]


]

J=1: A2’’: BO-ADIAJ=1: A2’’: BO-DIAB

logo


H+3 : Coupled PES: J=1, E ′- and E ′′-states (E ′′: 64.128 cm−1)

-5

-4

-3

-2

-1

0

1

2

3

2000 4000 6000 8000 10000 12000 14000 16000 18000

fre

qu

ency

dif

fere

nce

[cm

-1]

J=1: E’: BO-ADIAJ=1: E’: BO-DIAB

-6

-4

-2

0

2

4

6

0 2000 4000 6000 8000 10000 12000 14000 16000

fre

qu

ency

dif

fere

nce

[cm

-1]


]

J=1: E’’: BO-ADIAJ=1: E’’: BO-DIAB

logo


Conclusions

(1) It is possible to take care of non-adiabatic effects on the vibration-rotationspectra of small molecules, in terms of a single potential curve/surface and ageometry-dependent reduced mass, that differs for vibration and rotation.

It is obvious that non-adiabatic effects on vibration-rotation spectra (at leastthose of 2nd order) have mainly to do with the participation of the electrons inthe nuclear motion and hardly with a coupling of different electronic states.

(a) Good approximations for the non-adiabatic ansatz for a single PES can beobtained from rather simple models (LCAO-approach).(b) An energy expression is obtained correct to O(M−2), hence forhydrogen-containing molecules one can expect 6- or 7-figures accuracy.

(2) Numerical improvements: using sum-over-states formula (CPU-intensiv)(∆ν = 0.01 - 0.001 cm−1)(3) Strong non-adiabaticity: geometry-dependent eff. masses will not help→ Coupling different adiabatic PESsKinetic energy (adiabatic) versus Potential energy (diabatic) coupling

Acknowledgements: W. Kutzelnigg, M. Khoma

logo


H+3 : Deviations to experiment

additional analysis [26]. Search scans, fine scans, andtheoretical line centers are shown for some cases inFig. 1. The positions of the visible spectral lines are verywell reproduced by the most complete model (þ REL) ofthe present computations. Instead of exp�calc differencesof up to 3 cm�1 or even more displayed by the previouscalculations, the deviations are now significantly reducedwith a remaining rms deviation determined with respect tothe measured data being only �0:1 cm�1. This representsmore than an order of magnitude improvement over pre-vious ab initio computations and also a significant im-provement over semiempirical models [31] of thespectrum. In Fig. 2 we show the exp�calc differencesfor the present theory in the entire 0–16 600 cm�1 spectralregion and for J � 3 initial levels. Employing the

BOþ DA surface in the nuclear calculations producesro-vib energies whose deviations from the experimentsystematically increase from a very small number for thelowest transitions to about�1:5 cm�1 for the highest ones.However, when the results are corrected for the REL andNA effects, excellent agreement with the experiments isreached. The vibrational wave functions [26] of the inves-tigated states on the calculated PES show large probabilitydensity near linear configurations of the nuclei, thus dem-onstrating the significant role of such geometries in theinvestigated energy region.In conclusion, this work provides the most accurate

global ground-state Hþ3 PES available to date. Together

with a simple model for nonadiabatic effects, it has allowedus to predict the ro-vib transitions of Hþ

3 with unprece-

dented accuracy. By measurements extending far into thevisible region, our calculations are shown to match theobserved spectral lines with an average deviation as lowas 0:1 cm�1. The high predictive power of the presenttheory and the extreme experimental sensitivity achievedopen the door to further measurements of the Hþ

3 spectrum

at even higher energies. Moreover, the accuracy of thepresent global PES and its large spatial extent set thefoundation for describing and assigning the transitionsin the Carrington spectrum [3,4] which stretches into thedissociative continuum. The near-dissociation region of theHþ

3 PES is also important for understanding proton colli-

sions with hydrogen molecules. Finally, we note that PEScalculations with ECSGs, although computationally expen-sive, are not restricted to two-electron systems; they couldthus form a new paradigm for near-experimental-accuracy

0.1+5.0+5.0−0

1

2

4

3

0

2

4

6

15 058

15 058.680(5)

0

2

4

6

5.3+5.0+5.0− 15 450

0

2

10

6

4

8(b)

15 450.112(5)

16 506.139(5)

0

2

4

−3.0 −0.5 +1.0

0

2

10

6

4

8

16 506

Wave number (cm−1)

Sea

rch

scan

sig

nal (

arb.

uni

ts)

Fin

e sc

an s

igna

l (ar

b. u

nits

)

(a)

(c)

(d)16 660.240(5)

0

1

5

3

2

4

0

2

4

6

16 660−3.0 +0.5 +1.0 +8.5

NMTSAH03

NMTSAH03

(NMT)SAH03

(NMT)SAH03

FIG. 1 (color online). Visible-light spectroscopy of coldtrapped Hþ

3 ions for some of the measured lines detailed in

Table I. Top marks (full lines): present theoretical results.Orange (gray) symbols and fitted curves: fine scans (right-handvertical scales) with Doppler-broadened line profiles (effectiveion temperature �85 K). Dashed marks: earlier predictions byNMT [11] and SAH03 [16] as labeled (arrows and shadingindicate scale offsets). Black symbols: search scans (see thetext) with breaks marked by thin vertical lines (left-hand verticalscales).

0 5000 10000 15000

Eexp (cm−1)

−1.5

−1

−0.5

0

0.5

Eex

p −

Eca

lc (

cm−

1 )

FIG. 2 (color online). Differences between experimental tran-sition energies Eexp and the present calculation (Ecalc) for the

BOþ DA PES (diamonds) and for the BOþ DAþ REL PESplus NA corrections (asterisks). Small red asterisks andlight gray diamonds: using experimental data fromRefs. [12,28,30,32] (J � 3 initial levels); large orange asterisksand dark gray diamonds: present data with 000 ð1; 0Þ and ð1; 1Þinitial levels.

PRL 108, 023002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

13 JANUARY 2012

023002-4

Pavanello et al, PRL (2012)µvib=NU + 0.4753 me

µrot=NU

Theory: problem arises, when it comes to comparison with experiment,unless one is able to perform a full non-adiabatic calculation.The participation of the electrons in vibration and rotation appears in a ratherindirect way as a non-adiabatic effect.

logo


The ’vib-mass’ µ(r) (diatomic) in H+3

0

2

4

6

8

10

12 0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

me

R

r

me

0.50.7

0.5

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Contribution of the electron mass ∆me to the proton mass: µvibr (H2/H+

2 ) in H+3

0: nuclear mass, +1.0: atomic mass for two protons in H+3 .

logo


Quality of CI-R12, GG and PES-FIT

FCI-R12 with one linear r12 term in the wavefunction expansion

multi-reference FCI-R12 ansatz with several linear r12 terms in thewavefunction expansion (excited states)

absolute accuracy: <10−6 Hartree (<0.2 cm−1); >20000 energy points(regular grid)relative accuracy: even better (PES: Ei -Emin))

GG: some linear points not well converged (deleted from the fit): 4500energy points (MBB-grid)

RMS-values (in cm−1) for different fits of GG and CI-R12 energies with anenergy-cutoff Ecut and a maximum value for R(H-H2) (in Bohr).

RMS # points Ecut Rmax

GG-VAV-BO(global) 0.07 2560 37000 -GG-VAV-ETOT(global) 1.16 2560 37000 -GG-POLY-ETOT(local) 1.16 2404 - 6CIR12-F-POLY-ERANGE-BO(local) <0.01 ca. 8000 35000 -CIR12-F-VLABP-BO(global) 2.10 13693 75000 40CIR12-F-VLABP-POLYC-BO-ADIA(global) 0.01 13693 75000 40

logo


GGCIR12(f)

10000 cm-1 15000 cm-1 35000 cm-1

0 2 4 6 8 100

1

2

3

4

5

6

r(H

2)[

ao]

100 cm-1

150 cm-1

1000 cm-1

300 cm-1

115 cm-1

0 1 2 3 4 5 6 7 8

R(H-H2)[ao]

0.5

1

1.5

2

2.5

3

3.5

4

Local region around minimum and diagonal adiabatic corrections

logo


H2-molecule: adiabatic + non-adiabatic contributions

µnuc = NU BO ADIA N-ADIA1 ν (H2; 4161.14 cm−1) ∆ν: 2.2 ∆ν: 1.4 ∆ν: 0.82 ν (H2; 8086.93 cm−1) ∆ν: 4.2 ∆ν: 2.65 ∆ν: 1.55

3 ν (H2; 11782.36 cm−1) ∆ν: 5.8 ∆ν: 3.5 ∆ν: 1.3

logo


H+3 : Influence on the accuracy of the rovibrational spectrum

size of the contributions E to the potential and remaining errors of transitionfrequencies ∆ν (in cm−1).

various contributions (minimum region) ∆ν EBO electronic potential energy surface < 0.01 ≈ 40000Diagonal adiabatic corrections < 0.01 ≈ 115Relativistic contributions (Breit-Pauli) < 0.01 ≈ −2.6Nonadiab. corrections (1st and 2nd derivat.) ≈ 0.5 ≈ ±50Potential fit (local, global) ≈ 0.05, 1QED Lamb-shift 0 ≈ 0.27

Simulation of non-adiabatic effects (choice of the mass for nuclear motion)a) Nuclear mass (NU) or atomic mass (NU + 2

3 me for H+3 ) (NU23)

b) Atomic mass for vibrational motion and nuclear mass for rotational motion(VR, NU-Moss=NU + 0.4753 me) (Bunker und Moss, 1977, 1996))

logo


H+3 : Non-adiabatic shifts (Schiffels et al (2003))

14 2 Molecular Quantum Dynamicsì�íìíQî�íì ï¬ð5ñ*ò�óQô�õÁö�÷Yô�øKù_ú�óQô�û�ô�ügóQýNþ&öÁÿgÿ��Tý�ügóQö�÷�� "!#��$��&%��(')�* +��,�#��-�� +��/.0�1�2�+��.436587952:�;958795�<�=>��.��2��*%$��'@?A%��B��C��D��%��B��EC�F��"*�� + G��*�+��H�� I��8?J�2%��B��E��2��K�*'L��*��74MN.I�C�� 0�O��P��.Q �� +RS3T1�C��8�� U�;WVX:�Y�;Z:�[�\B=-;]��^ G�2?Z_]��E� ��2�+��@��* +��.��a`cbd �C��_��eEf��*�g�� 2��*% �.G��S�+��h��i&�2�� 2�+�/�L��+��,��j��NkP��*��Rml-neo�p(q�Vr5�s�;-52t2\67��h��*�� +�-��(i%��-�� G!u��%v +�2?6_8��E� ��F��C��C�� +��.P%��'L�� j�w +��!x��N�-y8�f�� +��!u�*�,�1��L��.z��.z.+�� Q �g��{ �� +�g58795 Q ��+�|�S!u�Fy8��!O �!}%��-_]�B�2�+��H��e[�~�5�!/�C��7��/%��*_]��2��$.0�� +��iH%��?

0

0.2

0.4

0.6

0.8

1

1.2

EC

alc

- EO

bs/c

m-1

(1,00)

(1,11)

(1,20)

(1,22)

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000

EC

alc

- EO

bs/c

m-1

ECalc/cm-1

(0,00)

(0,11)

(0,20)(0,22)

(0,31)

(0,33) A1

(0,33) A2

�� +�H587958��kP��!#�C�� +��.+��I�� 2��*%�3T�N��0��0=g��%m�-y8�f�� +��!u�*�,�1��i�%��*��* +!#��*%3��C�w�+=K +�F?Z_]��E� +�2��C��P��* +��.v�2��` bd VX:,Y8;*:�[*\67��* +�S��.u��i�� Q ��R� G��1�F��C�� G�� +�*.+.G��!OE��% Q ��+�&�h.+!u��a.+��F�+��* Q ��S��1�HE��%�36� �-� �,�� =�; Q ��I��%��-_8��2��.�� G�^!v�2��(iv%� ��+�K��.G�+ +��#_]��E� ��2�+��C�2�).+��.��/��EC��% �� G��.�7�f��%�.��h�+��¡_]��E� +�2��C��fEC��% �2��%v?D��"�K��.+.G�� y8�+��,��?¢��/ G��1�2�+��L��y8��2��W7

logo


Summary

The traditional formulation of a molecular wave function describing the motionof the electrons and the nuclei, in terms of the Born hierarchy,does not allow to account for the participation of the electrons in vibrational orrotational motions, except by recurring to a fully non-adiabatic treatment,abandoning completely the concept of a single potential energy surfacefor one electronic state of the molecule (’weak’ non-adiabaticity).

An alternative approach is based on an exact separation in all possibledissociation limits, and which leads to a Hamiltonian for the nuclear motionwith a geometry-dependent effective nuclear mass, that is different forvibration and rotation.

An energy expression is obtained correct to O(M−2) (large for hydrogen),where M is a representative nuclear mass in units of the electron mass.

Bunker und Moss (1977,1996), Essen (1977), Kutzelnigg (2007)

Non-adiabatic calculations for rovibrational states of H2...

Documents

Transcript of Non-adiabatic calculations for rovibrational states of H2...