MOTIVATION • PATH INTEGRAL APPROACH TO · 2013-01-04 · Department of Physics, TCOMP-EST Tapio Rantala: QUANTUM AND THERMAL MOTION IN MOLECULES FROM ... WSTC, 20 Dec 2012 PATH-INTEGRAL

Department of Physics, TCOMP-EST WSTC, 20 Dec 2012Tapio Rantala: QUANTUM AND THERMAL MOTION IN MOLECULES FROM ...

1QUANTUM AND THERMAL MOTION IN MOLECULES FROM FIRST-PRINCIPLES

Tapio T. Rantala,Department of Physics, Tampere University of Technology

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

• QUANTUM DYNAMICS AND• STATISTICAL PHYSICS USING• MONTE CARLO TECHNIQUE

• DEMONSTRATING THE FEATURES WITH H3+

• BEYOND BORN–OPPENHEIMER and ZERO-POINT MOTION • ELECTRON–NUCLEI COUPLING & ENERGETICS • CHEMICAL REACTIONS and RELATED ... • STRONG CORRELATION


MOTIVATION

Electronic structure is the key quantity to materials properties and related phenomena:Mechanical, thermal, electrical, optical,... .

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

electron correlations (CI, MCHF, DFT-functionals)• typically ignore nuclear quantum and thermal

dynamics and coupling of electron–nuclei dynamics (Born–Oppenheimer approximation)

• give the zero-Kelvin description, only.

2


PATH-INTEGRAL DESCRIPTION OF QUANTUM DYNAMICS

Time evolution of the wave function is laborious and ”challenging”:

• path sampling• interference of paths • . . .

Propagation in imaginary time is in better control. It can be used as formulation quantum statistical physics, and thus, finding the finite temperature electronic structure.

3

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)


CLASSICAL PATH

Let us consider particle dynamics from a to b.Lagrangian formulation of classical mechanics for finding the path/trajectory leads to equations of motion from minimization (extremum) of action

where the Lagrangian L = T–V.

=> =>

For example, the classical action of the free-particle is

4

S = 12m Δx

Δt"

#$

%

&'2

Δt = 12m (xb −xa )

2

(tb − ta ).

δS = 0ddt∂L∂x

−∂L∂x

= 0

a = (xa , ta )

b = (xb, tb )

Δx, Δt

S= L(x,x,t)dtta

tb

∫ ,

– ∂V∂x=mx


QUANTUM PATH

Usually, the most probable quantum path is the classical one, but other paths contribute, too, with a certain probability. Quantum probability of the particle propagation from a to b is P(b,a) = |K(b,a)|2 ,the absolute square of the probability amplitude K.The probability amplitude is the sum over all oscillating phase factors of the paths xab as

where the phase is proportional to the action

5

K(b,a) = φ[xab ]all xab

∑

φ[x(t)]= A×exp iS[x(t)]

#

$%

&

'(

φ

PARTICLE PHOTON

Class. Geom. mech. optics

Quantum Physicalmechanics optics

λ


PATH-INTEGRAL

Now, let us define the sum over all paths as a path-integral

We call this ”kernel” or ”propagator” or ”Green’s function”.In terms of stationary eigenstates it can be written as

The kernel satisfies the free-particle Schrödinger equation in space-time {xb,tb}.For example, the free-particle propagator takes now the form

6

K0(b,a) =m

2πi(tb − ta )#

$%

&

'(

1/2

exp im(xb −xa )2

2(tb − ta )#

$%

&

'(.

K(b,a) = e(i/ )S[b,a ]a

b

∫ Dx(t).

K(b,a) = φ*n (a)φn (b)n∑ e−( i/ )En ( tb−ta )


MIXED STATE DENSITY MATRIX

Considering all states of the particle, for the probability p(x) of finding the particle/system in configuration space at x, we have

Now, define the mixed state density matrix (in position presentation)

( )Thus, we find

and normalization implies Expectation values evaluated from

7

φn (x)

P(x) = pn (x)n∑ =

1Z

φ*n (x)φn (x)n∑ e−βEn .

ρ(x ',x) = φ*n (x ')φn (x)n∑ e−βEn .

Z = ρ(x,x)∫ dx = Tr(ρ).

P(x) = 1Zρ(x,x)

ρ(β) = e−βH.

β =1kT

A = Tr(ρA) / Z


PATH-INTEGRAL EVALUATION OF DENSITY MATRIX

Now, compare

and

in equilibrium (time independent hamiltonian) and tb > ta.Replacing (tb – ta) = u by –iℏβ or β = i(tb – ta)/ℏ (imaginary time period)

we obtain ρ(b,a), for which Cf.

for a time independent hamiltonian. Thus, we can evaluate the density matrix from a path-integral similarly

where the imaginary time action is

8

ρ(x ',x) = φ*n (x ')φn (x)n∑ e−βEn

∂ρ(b,a)∂β

= −Hb ρ(b,a).∂K(b,a)∂tb

= −iHb K(b,a)

K(b,a) = φ*n (a)φn (b)n∑ e−( i/ )En ( tb−ta )

S[x(u);β,0]= m2x2(u)+V(x(u))

!

"#$

%&0

b

∫ du.

ρ(xb,xa;β) = e(− i/ )S[β,0 ]all x(u )∫ Dx(u),


MONTE CARLO SAMPLING OFIMAGINARY TIME PATHS

For the density operator we can write

if the kinetic and potential energies in the hamiltonian H = T + Vcommute. This becomes exact at the limit of imaginary time period goes to zero, the high temperature limit, because the potential energy approaches constant in position representation for each imaginary time step.Thus, we can write

where , and M is called the Trotter number.This allows numerical sampling of the imaginary time paths with a Monte Carlo method.

9

ρ(β) = e−βH = e−β/2He−β/2H ,

ρ(r0 , rM;β) = ρ(r0 , r1; τ)ρ(r1, r2; τ) ...ρ(rM−1, rM; τ)∫∫∫ dr1dr2 ...drM−1,

τ =β /M β =1kT


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

10

Metropolis N. et al., J. Chem. Phys. 21, 1087, (1953).

Ceperley D.M., Rev. Mod. Phys. 67, 279, (1995) and in Monte Carlo and ... (Eds. K. Binder and G. Ciccotti, Editrice Compositori, Bologna, Italy 1996) Storer R.G., J. Math. Phys. 9, 964, (1968)


A-FEW-QUANTUM-PARTICLES SYSTEMS

• A COUPLE OF ELECTRONS IN QUANTUM DOTS• M. Leino & TTR, Physica Scripta T114, 44 (2004)• M. Leino & TTR, Few Body Systems 40, 237 (2007)

• HYDROGEN ATOMS ON Ni SURFACE• M. Leino, J. Nieminen & TTR, Surf. Sci. 600,1860 (2006)• M. Leino, I. Kylänpää & TTR, Surf. Sci. 601, 1246 (2007)

• ELECTRONS AND NUCLEI QUANTUM DYNAMICS• I. Kylänpää, M. Leino & TTR, PRA 76, 052508 (2007)• I. Kylänpää, TTR, J.Chem.Phys. 133, 044312 (2010)• I. Kylänpää, TTR, J.Chem.Phys. 135, 104310 (2011)

• THREE AND FOUR PARTICLE MOLECULES• I. Kylänpää, TTR, PRA 80, 024504, (2009)• I. Kylänpää, TTR and DM. Ceperley, PRA 86, 052506, (2012)

11


QUANTUM STATISTICAL PHYSICS PATH INTEGRAL MONTE CARLO APPROACH

An ab initio electronic structure approach with• FULL ACCOUNT OF CORRELATION, the

van der Waals interaction, for example!• TEMPERATURE DEPENDENCE• BEYOND BORN–OPPENHEIMER

APPROXIMATION• INTERPRETATION OF EQUILIBRIUM

DISSOCIATION REACTION

• Without the exchange interaction, now

12


SIZE AND TEMPERATURE SCALES

13

T / K

100 000

10 000

1 000RT

100

10

10 100 1000 10 000 100 000 size/atomsDFT

QCDFT

MOLDY

PIMC PATH INTEGRAL MONTE CARLO APPROACH

Ab initio-MOLDY


QUANTUM / CLASSICAL APPROACHES TO DYNAMICS

14

time dependent

T > 0equilibrium

T = 0

MOLECULAR

Wave packet approaches !

TDDFT

Car–Parrinello

and

DYNAMICS

Metropolis Monte Carlo Rovibrational

!ab initio MOLDY

Molecular mechanics

approaches

DMC

ab initio Quantum

Chemistry / DFT /

semiemp.

electronic dyn.:

nuclear dyn.:

Classical

Q

Q

Q

Q Q

Classical


H3+ MOLECULE

Quantum statistics 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.

15

e–

p+ p+

p+

e–

me, mp


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.

16


MOLECULAR GEOMETRY AT LOW TEMPERATURE: ZERO-POINT MOTION

Internuclear distance. Quantum nuclei, classical nuclei, with FWHM

17

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


PARTICLE–PARTICLE CORRELATIONS: ELECTRON–NUCLEI COUPLING

Pair correlation functions. Quantum p (solid), classical p (dashed) and Born–Oppenheimer (dash-dotted).

18

p–p

e–ep–p

p–e

0 5000 10000 15000

ï1.30

ï1.20

ï1.10

ï1.00

ï0.90

ï0.80

H3+

H2+H+

H2++H

2H+H+

Barrier To Linearity

T (K)


ENERGETICS AT HIGH TEMPERATURES: DISSOCIATION–RECOMBINATION

19

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


DISSOCIATION–RECOMBINATION EQUILIBRIUM REACTION

The molecule and its fragments:

20

The equilibrium composition of fragments at about 5000 K.

ï1.3 ï1.2 ï1.1 ï1.00

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!

Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

SOME THERMODYNAMICS

Expected total energy or internal energy is

and or ( )

All standard thermodynamic quantities and relations can be derived from the partition function Z or free energy F.

21

E =U =1Z

Ene−βEn

n∑ = kT2 ∂(lnZ)

∂T=∂(βF)∂β

.

∂F∂T

= −S

∂F∂V

= −P F =U−TS

dU = −PdV+dQ

dQ = TdS

Z = e−βEn

n∑ = e−βFE =U = −

1Z∂(Z)∂β

F = − 1βlnZ β =

1kT


PARTITION FUNCTION

22

Numerical integration of

gives the partition function. We use the boundary condition Z(0) = 1.

lnZ(T) = lnZ(T1)+EkBT

2T1

T

∫ dT


FREE ENERGY

23

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 H

artr

ee)

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

(un

its o

f k B

)

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

F = − 1βlnZ


ENTROPY

24

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 H

artr

ee)

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

(un

its o

f k B

)

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



MOLECULAR HEAT CAPACITY

25

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

(un

its o

f k B

)

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).

7T. T. Koskinen, A. D. Aylward, C. G. A. Smith, and S. Miller, Astrophys.J. 661, 515 (2007).

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,

2123 (1998).20M. F. Herman, E. J. Bruskin, and B. J. Berne, J. Chem. Phys. 76, 5150

(1982).21M. Misakian and J. C. Zorn, Phys. Rev. Lett. 27, 174 (1971).


CV =∂ E∂T


SOME FINAL NOTES AND THOUGHTS

• Dynamics is always present in molecules and all the constituent particles participate:

• in zero Kelvin and• in finite temperature

• In finite temperature• there are no stable molecules• classical dynamics emerge

26

Essay title: Ehrenfest theorem and decoherenceInstruction: Find definition/explanation of both from the literature and compare these two as a way from quantum mechanics to classical mechanics

�+

p+e–

e+

me, mp, m�

Documents

MOTIVATION • PATH INTEGRAL APPROACH TO · 2013-01-04 · Department of Physics, TCOMP-EST Tapio Rantala: QUANTUM AND THERMAL MOTION IN MOLECULES FROM ... WSTC, 20 Dec 2012 PATH-INTEGRAL