Non-adiabatic calculations for rovibrational states of H2...
Transcript of Non-adiabatic calculations for rovibrational states of H2...
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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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
Outline
1 Introduction
2 Distant-dependent effective masses
3 Coupled surfaces
4 Conclusion
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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).
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
New effective kinetic energy
Energy expectation value up to O(1/M2) (diatomic):
E =12〈 ∂χ∂Z| 1MR
(1 +AM
)| ∂χ∂Z〉+
12〈 ∂χ∂X| 1MR
(1 +BM
)| ∂χ∂X〉
+12〈 ∂χ∂Y| 1MR
(1 +BM
)| ∂χ∂Y〉+ 〈χ|Ead (Z )|χ〉 (8)
NAD corrections are included in distant-dependent mass-terms
Using the concept of a single PES
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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+)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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+)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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 )
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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)
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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.
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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?
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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]
transition frequency [cm-1
]
E: BO-ADIA E: BO-DIAB
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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]
transition frequency [cm-1
]
J=1: A2’’: BO-ADIAJ=1: A2’’: BO-DIAB
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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]
transition frequency [cm-1
]
J=1: E’’: BO-ADIAJ=1: E’’: BO-DIAB
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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.
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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 .
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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))
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
H+3 : Non-adiabatic shifts (Schiffels et al (2003))
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Introduction Distant-dependent effective masses Coupled surfaces Conclusion
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)