FEYNMAN PATH INTEGRAL APPROACH TO QUANTUM …butler.cc.tut.fi/.../QTMN-15.PI-appendix.pdfDepartment of Physics, TCOMP-EST Tapio Rantala: Feynman path integral approach to quantum dynamics

Department of Physics, TCOMP-EST 12.10.2015 Physics@PerthTapio Rantala: Feynman path integral approach to quantum dynamics

1FEYNMAN PATH INTEGRAL APPROACH TO QUANTUM DYNAMICS AND QUANTUM MONTE CARLO

Tapio Rantala, Tampere University of Technology, Finland

http://www.tut.fi/semiphys CONTENTS • MOTIVATION • FROM PATH INTEGRAL FORMALISM TO

• QUANTUM DYNAMICS and STATIONARY STATES • COHERENT DYNAMICS • BENEFITS OF MONTE CARLO TECHNIQUE • INCOHERENT DYNAMICS • TEST CASE: 1D HOOKE’S ATOM

• QUANTUM STATISTICAL PHYSICS APPROACH TO ELECTRONIC STRUCTURE THEORY • NOVEL FEATURES AND DEMONSTRATIONS


MOTIVATION

Electronic structure is the key concept in materials properties and related phenomena: for mechanical, thermal, electrical, optical, and also, for dynamics of electrons.

Conventional ab initio / first-principles type methods • suffer from laborious description of electronic

correlations (CI, MCHF, CC, DFT-functionals). • typically ignore nuclear (quantum) dynamics and

non-adiabatic coupling of electron–nuclei dynamics (Born–Oppenheimer approximation) —> ab initio MOLDY.

• give the zero-Kelvin description, only, though more realistic temperatures might be relevant.

• are typically good for stationary states, only.

2


PATH INTEGRAL APPROACH TO QUANTUM MECHANICS

3

H (x, t) = i@ (x, t)

@t

K(x

b

, t

b

;x

a

, t

a

) =

Zxb

xa

exp(iS[x

b

, x

a

])Dx(t)

(x, t) =

ZK(x, t;xa, ta) (xa, ta)dxa

(x, t) =

1X

n=0

Cn�n(x) exp[�iEnt]

Real Time Path Integrals (RTPI): Time evolution in real time

Stationary states from incoherent RTPI

Re

Im

τ: 0 → -iℏβ

t

τ



4

H (x, t) = i@ (x, t)

@t⌧ = it

(x, ⌧) =

1X

n=0

Cn�n(x) exp[�En⌧ ]

(x, ⌧) =

ZG(x, ⌧ ;xa, ⌧a) (xa, ⌧a)dxa

K(x

b

, t

b

;x

a

, t

a

) =

Zxb

xa

exp(iS[x

b

, x

a

])Dx(t)

(x, t) =


(x, t) =

1X

n=0

Cn�n(x) exp[�iEnt]

Real Time Path Integrals (RTPI): Time evolution in real time

Quantum statistical physics in equilibrium at temperature T, τb-τa =�= 1/kBT

Stationary states from incoherent RTPI Diffusion Monte Carlo (DMC)

H (x, ⌧) = �@ (x, ⌧)@⌧

⇢(x

b

, ⌧

b

;x

a

, ⌧

a

) =

Zxb

xa

exp(�S[x

b

, x

a

])Dx(⌧)



There is nothing but dynamics and all possible paths contribute with a complex phase. Path integral (sum over all paths) gives the probability amplitude, the wave function.

in real time in imaginary time ( t ) ( t = -iℏβ )

Feynman–Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965) Feynman R.P., Rev. Mod. Phys. 20, 367–387 (1948) Feynman R.P., Statistical Mechanics (Westview, Advance Book Classics, 1972)

5

Time evolution in real time & stationary states

(RTPI)

Quantum statistical physics in equilibrium with

temperature T, � = 1/kBT


REAL TIME PATH INTEGRAL APPROACH TO COHERENT QUANTUM DYNAMICS

6

H (x, t) = i@ (x, t)

@t

K(x

b

, t

b

;x

a

, t

a

) =

Zxb

xa

exp(iS[x

b

, x

a

])Dx(t)

(x, t) =


K(xb, xa;�t) ⇡h

m

2⇡i~�t

iD/2exp

i

~ (m

2�t

(xb � xa)2 � �t

2

(V (xa) + V (xb))

�

Glauber state evolution in a regular grid with different kernels and time steps.Short time kernel or Trotter kernel

Monte Carlo grid to: - avoid artificial interferences - sample the space continuously - allow adaptation along time evolution – the walkers

and walker distribution

I. Ruokosenmäki and TTR, Numerical path integral approach to quantum dynamics and stationary states, Communications in Computational Physics 18, 91 (15).


REAL TIME PATH INTEGRAL APPROACH TO INCOHERENT PROPAGATION

7

(x,�t) =

1X

n=0

Cn�n(x) exp[�i(En � ET )�t]

(x,�t) ⇡1X

n=0

Cn�n(x){1� [(En � ET )�t]2/2� i[(En � ET )�t]}

R(x,�t) =1X

n=0

Cn�n(x){1� [(En � ET )�t]2/2}

(x, t) =


with small angle approximation

Coherent propagation and zero reference shift

Incoherent propagation: ignore the very small imaginary part

This is simulation of quantum Zeno effect as ∆t → 0, i.e., keeping the system in its observable real state – or finding the eigenstate closest to ET , the same way as DMC finds the ground state. Cool, isn’t it!

EL(x) = ��(x)

�t

��(x)

Re

Im

(x,�t)


ONE DIMENSIONAL HOOKE’S ATOM: SEPARATION OF COORDINATES

8

r

H(x1, x2) = �1

2r2

1 �1

2r2

2 +1

2!2x2

1 +1

2!2x2

2 +1

|x1 � x2|H(r,R) = � 1

2µr2

r +1

2µ!2r2 +

1

|r| �1

2Mr2

R +1

2M!2R2 ⌘ Hr +HR

-10 -5 5 10r

-0.4

-0.2

0.2

0.4

u0(r )


1D HOOKE’S ATOM: GROUND STATE WAVE FUNCTION

9

r


1D HOOKE’S ATOM: PHASE OF THE WAVE FUNCTION

10


1D HOOKE’S ATOM: RTPI AND DMC

11

I. Ruokosenmäki et al., Numerical path integral solution to strong Coulomb correlation in one dimensional Hooke’s atom, submitted to CPC, (arXiv:1510.02230).


PATH INTEGRALS IN IMAGINARY TIME

Feynman path integral in imaginary time evaluates the mixed state density matrix

The partition function is obtained as

where � = 1 / kBT

and S[R] is the action of the path R(0) --> R(�) for the imaginary time period �.

12

Note! There is no wave function!


NVT ENSENBLE QUANTUM STATISTICAL MECHANICS

Thus, we have the quantum statistical mechanics and the concepts of NVT ensemble

• finite temperature, T > 0 • density matrix

action • partition function • expectation values

• Note! this is all exact, so far!

13

Feynman–Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965) Feynman R.P., Statistical Mechanics (Westview, Advance Book Classics, 1972)


DISCRETIZING THE PATH ...

With the primitive approximation and Trotter expansion

where M is the Trotter number and With short enough τ = β/M the action

approaches that of free particle as the potential function .

14

Note! This is not sc. PIMD


... FOR EVALUATION WITH MONTE CARLO

Monte Carlo allows straightforward numerical procedure for evaluation of multidimensional integrals.

Metropolis Monte Carlo • NVT (equilibrium) ensemble • now yields mixed state density matrix with almost classical transparency

15

Metropolis N. et al., J. Chem. Phys. 21, 1087, (1953). Ceperley D.M., Rev. Mod. Phys. 67, 279, (1995)


PATH INTEGRAL MONTE CARLO (PIMC)

An ab initio electronic structure approach withnovel features: • FIXED FINITE TEMPERATURE, NVT ENSEMBLE• NO BORN–OPPENHEIMER APPROXIMATION• FULL ACCOUNT OF CORRELATION,

the van der Waals interaction, for example!• Evaluation of EXCHANGE INTERACTION suffers

from Fermion Sign Problem!• SIMULATION OF EQUILIBRIUM DISSOCIATION–

RECOMBINATION REACTION

16


QUANTUM DOTS IN QCA

• Finite temperature and electron delocalization effects

17


SMALL LIGHT NUCLEI MOLECULES: ELECTRONS, PROTONS AND POSITRONS

3-particle and 4-particle ”molecules”

18

H– e+ or PsH

Ps2

H2

@ 300 K

Kylänpää, TTR and D.M. Ceperley, PRA 86, 052506 (2012)

H–

H2+

Ps2+, Ps–

e+H


DIPOSITRONIUM Ps2

• Pair approximation and matrix squaring

• Bisection moves • Virial estimator for the kinetic energy

• same average ”time step” for all temperatures ‹ τ › = β/M = 0.015, M ≈ 105

M = 217 ... 214

e–

e–

e+

e+

19


H3+ MOLECULE

Quantum statistical physics of two electrons and three nuclei (five-particle system) as a function of temperature:

• Structure and energetics: • quantum nature of nuclei • pair correlation functions,

contact densities, ... • dissociation temperature

• Comparison to the data from conventional quantum chemistry.

20

e–

p+ p+

p+

e–

me, mp

Ab initio or first-principles!


H3+ TOTAL ENERGY:

FINITE NUCLEAR MASS AND ZERO-POINT ENERGY

Total energy of the H3+ ion up to the dissociation temperature. Born–Oppenheimer approximation, classical nuclei and quantum nuclei.

21


H3+ MOLECULAR GEOMETRY AT

LOW TEMPERATURE: ZERO-POINT MOTION

Internuclear distance. Quantum nuclei, classical nuclei, with FWHM

22

Snapshot from simulation, projection to xy-plane. Trotter number 216.

0 5000 10000 15000

−1.30

−1.20

−1.10

−1.00

−0.90

−0.80

H3+

H2+H+

H2++H

2H+H+

Barrier To Linearity

T (K)


H3+ ENERGETICS AT HIGH

TEMPERATURES

23

Total energy of the H3+ beyond dissociation temperature. Lowest density, mid density, highest density.

0 5000 10000 15000

−1.30

−1.20

−1.10

−1.00

−0.90

−0.80

H3+

H2+H+

H2++H

2H+H+

Barrier To Linearity

T (K)


H3+ ENERGETICS AT HIGH

TEMPERATURES: DISSOCIATION–RECOMBINATION

24

Total energy of the H3+ beyond dissociation temperature. Lowest density, mid density, highest density.


DISSOCIATION–RECOMBINATION EQUILIBRIUM REACTION

The molecule and its fragments:

25

The equilibrium composition of fragments at about 5000 K.

−1.3 −1.2 −1.1 −1.00

20

40

60

80

100

120

140

160 H3+ H2+H+ H2

++H 2H+H+

BTL

E (units of Hartree)

Num

ber o

f Mon

te C

arlo

Blo

cks

We also evaluate molecular free energy, entropy and heat capacity!


PARTITION FUNCTION26

Numerical integration of

gives the partition function. We use the initial condition Z(0) = 0.

lnZ(T) = lnZ(T1)+EkBT

2T1

T

∫ dT


FREE ENERGY27

104310-5 Dissociation–recombination equilibrium J. Chem. Phys. 135, 104310 (2011)

The weighted least squares fit of the above energy func-tion, Eq. (8), to our data for temperatures up to about 3900 K,see Table I, gives the parameters,

a = 0.00157426,

b = 0.000132273,

c = −6.15622 × 10−6,

d = 0.00157430,

α = 269.410, and

D = a/b ≈ 11.9016.

In the fit, in addition to the (2SEM)−2 weights, we force thefirst derivative of the energy with respect to the temperature tobe monotonically increasing up to 3900 K. The fit extrapolatesthe 0 K energy to about 0.000 549EH above that of the para-H+

3 , i.e., it gives an excellent match within the statistical errorestimate.

In Fig. 3, the function ln Z(T ) from Eq. (9) is shown inthe range 0 < T < 4000 K — the behavior of the model athigher T is illustrated by the dashed line. Above 4000 K, thethree curves for different densities are obtained from thoseshown in Fig. 1 by numerical integration of Eq. (7) as

ln Z(T ) = ln Z(T1) +! T

T1

⟨E⟩kBT 2

dT , (10)

where T1 = 500 K.In Ref. 2, Neale and Tennyson (NT) have presented the

partition function ln Z(T ) based on a semi-empirical poten-tial energy surface, see Fig. 3. The NT partition function hasconventionally been used in atmospheric models. The overallshape is similar to the one of ours. However, the energy ⟨E⟩evaluated from their fit tends to be systematically lower thanours, although roughly within our 2SEM error limits. Thus,the deviations are not visible in Fig. 1. The energy zero of theNT fit, black dots in Figs. 1 and 3, is the same as ours in thiswork, and thus, allows direct comparison in Fig. 1.

Also in Fig. 3, the difference due to the choice of theJ = 0 state as the zero reference is illustrated by the NT par-tition function values, black pluses — the shape is notably af-

0 2000 4000 6000 8000 100000

5

10

15

20

T (K)

ln(Z

)

FIG. 3. The molecular NV T ensemble ln Z(T ) from the energetics in Fig. 1with the same notations. The blue solid line below 4000 K and its extrapo-lation (dashed line) are from Eq. (9), whereas the curves for three densitiesare from Eq. (10). The ln Z(T ) data (black pluses) and the fit (black dots)of Ref. 2 are also shown. The black dots have the same zero energy as thepartition function of this work (see text).

0 2000 4000 6000 8000 10000−0.8

−0.6

−0.4

−0.2

0

T (K)

Free

ene

rgy

(uni

ts o

f Har

tree

)

FIG. 4. Helmholtz free energy from Eq. (5) in the units of Hartree. Notationsare the same as in Fig. 3.

fected at low T , only. As mentioned above, already, the zeroreference of ln Z can be chosen differently.

Our low temperature partition function, Eq. (9), is closeto complete. With the PIMC approach, we implicitly includeall of the quantum states in the system with correct weightwithout any approximations. This partition function is the bestone for the modeling of the low density H+

3 ion containing at-mospheres, at the moment. However, it is valid up to the disso-ciation temperature, only. As soon as the density dependencestarts playing larger role, more complex models are needed.Such models can be fitted to our PIMC data given in Tables Iand II.

C. Other thermodynamic functions

In Fig. 4, we show the Helmholtz free energy from com-bined Eqs. (6) and (9). As expected, lower density or largervolume per molecule lowers the free energy due to the in-creasing entropic factor. Dissociation and the consequentfragments help in filling both the space and phase space moreuniformly or in less localized manner.

This kind of decreasing order is seen more clearly in theincreasing entropy, shown in Fig. 5. The entropy has beenevaluated from

S = U − F

T, (11)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

T (K)

Ent

ropy

(uni

ts o

f kB)

FIG. 5. Entropy from Eq. (11) in the units of kB. Notations are the same asin Fig. 3.

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp


ENTROPY28

104310-5 Dissociation–recombination equilibrium J. Chem. Phys. 135, 104310 (2011)

The weighted least squares fit of the above energy func-tion, Eq. (8), to our data for temperatures up to about 3900 K,see Table I, gives the parameters,

a = 0.00157426,

b = 0.000132273,

c = −6.15622 × 10−6,

d = 0.00157430,

α = 269.410, and

D = a/b ≈ 11.9016.

In the fit, in addition to the (2SEM)−2 weights, we force thefirst derivative of the energy with respect to the temperature tobe monotonically increasing up to 3900 K. The fit extrapolatesthe 0 K energy to about 0.000 549EH above that of the para-H+

3 , i.e., it gives an excellent match within the statistical errorestimate.

In Fig. 3, the function ln Z(T ) from Eq. (9) is shown inthe range 0 < T < 4000 K — the behavior of the model athigher T is illustrated by the dashed line. Above 4000 K, thethree curves for different densities are obtained from thoseshown in Fig. 1 by numerical integration of Eq. (7) as

ln Z(T ) = ln Z(T1) +! T

T1

⟨E⟩kBT 2

dT , (10)

where T1 = 500 K.In Ref. 2, Neale and Tennyson (NT) have presented the

partition function ln Z(T ) based on a semi-empirical poten-tial energy surface, see Fig. 3. The NT partition function hasconventionally been used in atmospheric models. The overallshape is similar to the one of ours. However, the energy ⟨E⟩evaluated from their fit tends to be systematically lower thanours, although roughly within our 2SEM error limits. Thus,the deviations are not visible in Fig. 1. The energy zero of theNT fit, black dots in Figs. 1 and 3, is the same as ours in thiswork, and thus, allows direct comparison in Fig. 1.

Also in Fig. 3, the difference due to the choice of theJ = 0 state as the zero reference is illustrated by the NT par-tition function values, black pluses — the shape is notably af-

0 2000 4000 6000 8000 100000

5

10

15

20

T (K)

ln(Z

)

FIG. 3. The molecular NV T ensemble ln Z(T ) from the energetics in Fig. 1with the same notations. The blue solid line below 4000 K and its extrapo-lation (dashed line) are from Eq. (9), whereas the curves for three densitiesare from Eq. (10). The ln Z(T ) data (black pluses) and the fit (black dots)of Ref. 2 are also shown. The black dots have the same zero energy as thepartition function of this work (see text).

0 2000 4000 6000 8000 10000−0.8

−0.6

−0.4

−0.2

0

T (K)

Free

ene

rgy

(uni

ts o

f Har

tree

)

FIG. 4. Helmholtz free energy from Eq. (5) in the units of Hartree. Notationsare the same as in Fig. 3.

fected at low T , only. As mentioned above, already, the zeroreference of ln Z can be chosen differently.

Our low temperature partition function, Eq. (9), is closeto complete. With the PIMC approach, we implicitly includeall of the quantum states in the system with correct weightwithout any approximations. This partition function is the bestone for the modeling of the low density H+

3 ion containing at-mospheres, at the moment. However, it is valid up to the disso-ciation temperature, only. As soon as the density dependencestarts playing larger role, more complex models are needed.Such models can be fitted to our PIMC data given in Tables Iand II.

C. Other thermodynamic functions

In Fig. 4, we show the Helmholtz free energy from com-bined Eqs. (6) and (9). As expected, lower density or largervolume per molecule lowers the free energy due to the in-creasing entropic factor. Dissociation and the consequentfragments help in filling both the space and phase space moreuniformly or in less localized manner.

This kind of decreasing order is seen more clearly in theincreasing entropy, shown in Fig. 5. The entropy has beenevaluated from

S = U − F

T, (11)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

T (K)

Ent

ropy

(uni

ts o

f kB)

FIG. 5. Entropy from Eq. (11) in the units of kB. Notations are the same asin Fig. 3.



MOLECULAR HEAT CAPACITY29

104310-6 I. Kylänpää and T. T. Rantala J. Chem. Phys. 135, 104310 (2011)

0 500 1000 1500 2000 2500 3000 35000

3/2

6/2

9/2

Rotation

Vibration

T (K)

Hea

t cap

acity

(uni

ts o

f kB)

FIG. 6. Molecular heat capacity as a function of temperature calculated usingthe analytical model of this work. The values on the y-axis are given in unitsof the Boltzmann constant kB.

where the internal energy is U = ⟨E⟩ − ⟨E⟩T =0. As expected,both the total energy (internal energy) and entropy reveal thedissociation taking place, similarly.

Finally, in Fig. 6, we present the molecular constant vol-ume heat capacity

CV = ∂⟨E⟩∂T

, (12)

where ⟨E⟩ is taken from Eq. (8), which is valid below disso-ciation temperatures, only.

Considering the goodness of our functional form for ⟨E⟩,it is very convincing to see the plateau at about 3/2kB corre-sponding to “saturation” of the contribution from the three ro-tational degrees of freedom. Thus, above 200 K the rotationaldegrees of freedom obey the classical equipartition principleof energy. It is the last term in the functional form of Eq. (8),that gives the flexibility for such detailed description of theenergetics.

It should be emphasized that the plateau is not artificiallyconstructed to appear at 3/2kB, except for a restriction givenfor the first derivative of the total energy to be increasing.Thus, the analytical model we present, Eq. (8), is found tobe exceptionally successful at low temperatures, i.e., belowdissociation temperature.

IV. CONCLUSIONS

We have evaluated the temperature dependent quantumstatistics of the five-particle molecular ion H+

3 at low densi-ties far beyond its apparent dissociation temperature at about4000 K. This is done with the PIMC method, which is ba-sis set and trial wavefunction free approach and includes theCoulomb interactions exactly. Thus, we are able to extendthe traditional ab initio quantum chemistry with full accountof correlations to finite temperatures without approximations,also including the contributions from nuclear thermal andequilibrium quantum dynamics.

At higher temperatures, the temperature dependentmixed state description of the H+

3 ion, the density dependentequilibrium dissociation–recombination balance, and the en-

ergetics have been evaluated for the first time. With the risingtemperature the rovibrational excitations contribute to the en-ergetics, as expected, whereas the electronic part remains inits ground state in the spirit of the Born–Oppenheimer ap-proximation. At about 4000 K the fragments of the molecule,H2 + H+, H+

2 + H, and 2H + H+, start contributing. There-fore, presence of the H+

3 ion becomes less dominant and even-tually negligible in high enough T .

We have also shown how the partial decoherence in themixed state can be used for interpretation of the fragmentcomposition of the equilibrium reaction. Furthermore, wehave evaluated explicitly the related molecular partition func-tion, free energy, entropy, and heat capacity, all as functionsof temperature. An accurate analytical functional form for theinternal energy is given below dissociation temperature. Weconsider all these as major additions to the earlier publishedstudies of H+

3 , where the dissociation–recombination reactionhas been neglected.

It is fair to admit, however, that PIMC is computationallyheavy for good statistical accuracy and approximations areneeded to solve the “Fermion sign problem” in cases whereexchange interaction becomes essential. With H+

3 , however,we do not face the Fermion sign problem, as the proton wave-functions do not overlap noteworthy and the two electronscan be assumed to form a singlet state, due to large singletto triplet excitation energy.

ACKNOWLEDGMENTS

We thank the Academy of Finland for financial support,and we also thank the Finnish IT Center for Science (CSC)and Material Sciences National Grid Infrastructure (M-grid,akaatti) for computational resources.

1J. J. Thomson, Phil. Mag. 21, 225 (1911).2L. Neale and J. Tennyson, Astrophys. J. 454, L169 (1995).3M. B. Lystrup, S. Miller, N. D. Russo, J. R. J. Vervack, and T. Stallard,Astrophys. J. 677, 790 (2008).

4T. T. Koskinen, A. D. Aylward, and S. Miller, Astrophys. J. 693, 868(2009).

5L. Neale, S. Miller, and J. Tennyson, Astrophys. J. 464, 516 (1996).6G. J. Harris, A. E. Lynas-Gray, S. Miller, and J. Tennyson, Astrophys. J.600, 1025 (2004).

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8B. M. Dinelli, O. L. Polyansky, and J. Tennyson, J. Chem. Phys. 103, 10433(1995).

9D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).10M. Pierce and E. Manousakis, Phys. Rev. B 59, 3802 (1999).11Y. Kwon and K. B. Whaley, Phys. Rev. Lett. 83, 4108(4) (1999).12L. Knoll and D. Marx, Eur. Phys. J. D 10, 353 (2000).13J. E. Cuervo and P.-N. Roy, J. Chem. Phys. 125, 124314 (2006).14I. Kylänpää and T. T. Rantala, Phys. Rev. A 80, 024504 (2009).15I. Kylänpää and T. T. Rantala, J. Chem. Phys. 133, 044312 (2010).16R. P. Feynman, Statistical Mechanics (Perseus Books, Reading, MA,

1998).17R. G. Storer, J. Math. Phys. 9, 964 (1968).18N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and

E. Teller, J. Chem. Phys. 21, 1087 (1953).19C. Chakravarty, M. C. Gordillo, and D. M. Ceperley, J. Chem. Phys. 109,

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THANK YOU!30

For more details see http://www.tut.fi/semiphys1. Ilkka Kylänpää, PhD Thesis (Tampere University of Technology, 2011)2. Ilkka Kylänpää and Tapio T. Rantala, Phys. Rev. A 80, 024504, (2009) and I. Kylänpää, M. Leino and T.T. Rantala, Phys. Rev. A 76, 052508, (2007)3. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 133, 044312 (2010)4. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 135, 104310 (2011)5. Ilkka Kylänpää, Tapio T. Rantala, David M. Ceperley, Phys. Rev. A 86, 052506, (2012) ACKNOWLEDGEMENTS Ilkka Ruokosenmäki, Ilkka Kylänpää, Markku Leino, Juha Tiihonen (TUT) Ilkka Kylänpää, David M. Ceperley (UIUC)

H3+

PS2

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FEYNMAN PATH INTEGRAL APPROACH TO QUANTUM …butler.cc.tut.fi/.../QTMN-15.PI-appendix.pdfDepartment of Physics, TCOMP-EST Tapio Rantala: Feynman path integral approach to quantum dynamics