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Department of Physics, TCOMP-EST NGS-NANO 3 Oct 2011Tapio Rantala: FROM QUANTUM STATISTICS TO QUANTUM CHEMISTRY

1FROM QUANTUM STATISTICAL PHYSICSTO FINITE TEMPERATURE

QUANTUM CHEMISTRY

Tapio Rantala, Department of Physics, TUT http://www.tut.fi/semiphys

CONTENTS • INTRODUCTION • MOTIVATION

• QUANTUM STATISTICS FOR • ELECTRONS -> FINITE TEMP.

QUANTUM CHEMISTRY • ATOMS -> QUANTUM DYNAMICS • ELECTRONS & ATOMS ->

MORE EXOTIC CHEMISTRY & PHYSICS


THEORETICAL AND COMPUTATIONAL MATERIALS PHYSICS – ELECTRONIC STRUCTURE THEORY

Compound semiconductor physics

Surface science

(Bio)molecular

(opto)electronics

Path-Integral Quantum Monte Carlo

CBM

VBM LUMO

HOMO

(100)

(110)

2


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-funtionals)• typically ignore nuclear (quantum) dynamics and

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

• give the zero-Kelvin description, only. So, how about the effects from finite temperature?

3

Electronic Structure Theory

ww

w.t

ut.f

i/sem

iphy

s

SiO2

!

The last few years

Temperature or TimeDependent States

Quantum Statistics! Feynman path-integral approach

Path Integral Monte Carlo (PIMC)! Explicitly temperature dependent NVT-ensemble simulation! Full account of electronic correlations! Allows model and calculations beyond Born–Oppenheimer approximation,

i.e., potential energy surface (PES)! Accepting the challenge of

FERMION SIGN PROBLEM !

Research FieldsMany-body Physics of Electrons! Quantum dynamics of light–matter interaction! Development of Quantum Monte Carlo approach! Development of Density Functional Theory! Finite temperature quantum statistics

Materials Physics and Chemistry! Compound semiconductors: bulk defects, surfaces, interfaces, nanostructures! Organic materials, photoabsorption, electron transfer, nanostructures! Finite temperature Quantum Chemistry

Eigenvalue Problem! At zero-Kelvin

! Hartree–Fock, CI, MCHF! Density Functional Theory(DFT – ”the work horse”)

Ground State andExcited States

The Present and The Challenges

Explicit Time Propagation! At zero-Kelvin! Time-Dependent DFT (TDDFT)

Electronic structure is the basic concept behind structure and properties of matter!

Robust Solution to the Semiconductor BAND GAP PROBLEM

! Evaluation of the missing discontinuity: Total (solid curve), LDA–KS (square) and GGA-KS (circle)compared with the experimental (star).

! Further work ongoing

M. Leino: PhD Thesis (2007) M. Mäki-Jaskari: PhD Thesis (2004)

O. Cramariuc: PhD Thesis (2006)H-P. Komsa: PhD Thesis (2008)

Surface physics and chemistry ! Adsorption of small and aromatic molecules! Adsorbate (quantum) dynamics! Van der Waals interaction! Magnetic defects

Compound semiconductor physics! Surface and interface electronic structure: bands and charge carriers! X-STM image simulation! Defect structure and (formation) energies! Optical properties (light–matter interaction)

Applications with DFT Small molecules and atoms! H2

+, He, H–, H2, Ps2, H3+ H2+H+ H2

++H, so far! Finite temperature included! Exact correlation, nodal surfaces for exchange

Continuous/periodic systems! Two-component Fermi sea: e– p+, e– e+

! LDA type correlation functional for the above

Ab initio chemical reactions! Dissociation–recombination in thermal equilibr.! Model for on or beyond BO-PES! Work for time dependence in progress! Fundamental issues in classical to quantum transition: decoherence

T>0 Ab Initio Quantum Chemistry


ELECTRONIC STRUCTURE THEORY

http://www.tut.fi/semiphys

4


SIZE AND TEMPERATURE SCALES

5

T / K

100 000

10 000

1 000RT

100

10

10 100 1000 10 000 100 000 size/atomsDFT

QCDFT

MOLDYAb initio-MOLDY


QUANTUM / CLASSICAL APPROACHES TO DYNAMICS

6

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 ab initio Quantum

Chemistry / DFT /

semiemp.

electronic dyn.:

nuclear dyn.:

Class

Q

Q

Q

Q Q

Class


PATH-INTEGRAL DESCRIPTION OF QUANTUM MECHANICS

Postulate:• All possible paths are relevant• Probability amplitude with interference

(wave function and uncertainty)• Action dependent phase factor (particle–

wave dualism)

–> Wave function and operator QMandtime-dependent dynamics in real time or ...

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

7


PATH-INTEGRAL DESCRIPTION OF IMAGINARY TIME DYAMICS ...

... or statistics in imaginary time ( t = -iℏβ ) for

• finite temperature, T > 0• density matrix

action:• partition function• expectation values

• including exact many-body interactions !

8

Based on the quantum statistics, from the mixed state density matrix (path-integral)

the partition function is obtained as

where! ! ! ! ! ! β = 1 / kBTand S(R) is the action of the path R(τ) in imaginary time τ.


... LEADS TO QUANTUM STATISTICS9


DISCRETIZE THE PATH ...

using Feynman path-integral

where M is the Trotter number andWith short enough τ = β/Mthe action

approaches that of free particleas U τ .

10


... FOR EVALUATION WITH MONTE CARLO

MC allows straightforward numerical procedure for evaluation of multidimensional integrals.

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

11


SIZE AND TEMPERATURE SCALES

12

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

13

time dependent

T > 0equilibrium

T = 0

MOLECULAR

Wave packet approaches

REAL-TIME PATHS

TDDFT

Car–Parrinello

and

DYNAMICS

Metropolis Monte Carlo Rovibrational

PIMCab initio MOLDY

Molecular mechanics

approaches ab initio Quantum

Chemistry / DFT /

semiemp.

electronic dyn.:

nuclear dyn.:

Class

Q

Q

Q

Q Q

Class

Department of Physics, TCOMP 24 Nov 2010Tapio Rantala: TUT COMPUTATIONAL PHYSICS SEMINAR: TCOMP

QUANTUM DOTS, ATOMS AND SMALL MOLECULES

• A COUPLE OF ELECTRONS IN QUANTUM DOTS

• M. Leino & TTR, Physica Scripta T114, 44 (2004)

• M. Leino & TTR, Few Body Systems 40, 237 (2007)

• ELECTRON AND PROTON QUANTUM DYNAMICS

• I. Kylänpää, M. Leino & TTR, PRA 76, 052508 (2007)

• THERMAL DISSOCIATION OF DIPOSITRONIUM

• Cassidy & Mills Jr.: Exp. obs. of Ps2, Nature 449, 195 (2007)

• I. Kylänpää & TTR, PRA 80, 024504 (2009)

e–

e–

e+

e+

e–

p+ p+

M. Leino: PhD Thesis (2007)

14


CASE STUDY: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–


ENERGETICS AT ZERO KELVIN LIMIT

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

16


PARTICLE–PARTICLE CORRELATIONS

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

17

p–p

e–ep–p

p–e


MOLECULAR GEOMETRYAT ZERO KELVIN LIMIT

Internuclear distance. Quantum nuclei, classical nuclei, with FWHM of distrib.

18

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)


ENERGETICS AT HIGHER TEMPERATURES

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.

1.3 1.2 1.1 1.00

20

40

60

80

100

120

140

160 H3+ H

2+H+ H

2++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!


PATH INTEGRAL QUANTUMMONTE CARLO APPROACH

An ab initio electronic structure approach with• FULL ACCOUNT OF CORRELATION• TEMPERATURE DEPENDENCE• BEYOND BORN–OPPENHEIMER

APPROXIMATION• INTERPRETATION OF EQUILIBRIUM

DISSOCIATION REACTION

• So far, without the exchange interaction

21

Teksti

Electronic Structure Theory

ww

w.t

ut.f

i/sem

iphy

s

SiO2

!

The last few years

Temperature or TimeDependent States

Quantum Statistics! Feynman path-integral approach

Path Integral Monte Carlo (PIMC)! Explicitly temperature dependent NVT-ensemble simulation! Full account of electronic correlations! Allows model and calculations beyond Born–Oppenheimer approximation,

i.e., potential energy surface (PES)! Accepting the challenge of

FERMION SIGN PROBLEM !

Research FieldsMany-body Physics of Electrons! Quantum dynamics of light–matter interaction! Development of Quantum Monte Carlo approach! Development of Density Functional Theory! Finite temperature quantum statistics

Materials Physics and Chemistry! Compound semiconductors: bulk defects, surfaces, interfaces, nanostructures! Organic materials, photoabsorption, electron transfer, nanostructures! Finite temperature Quantum Chemistry

Eigenvalue Problem! At zero-Kelvin

! Hartree–Fock, CI, MCHF! Density Functional Theory(DFT – ”the work horse”)

Ground State andExcited States

The Present and The Challenges

Explicit Time Propagation! At zero-Kelvin! Time-Dependent DFT (TDDFT)

Electronic structure is the basic concept behind structure and properties of matter!

Robust Solution to the Semiconductor BAND GAP PROBLEM

! Evaluation of the missing discontinuity: Total (solid curve), LDA–KS (square) and GGA-KS (circle)compared with the experimental (star).

! Further work ongoing

M. Leino: PhD Thesis (2007) M. Mäki-Jaskari: PhD Thesis (2004)

O. Cramariuc: PhD Thesis (2006)H-P. Komsa: PhD Thesis (2008)

Surface physics and chemistry ! Adsorption of small and aromatic molecules! Adsorbate (quantum) dynamics! Van der Waals interaction! Magnetic defects

Compound semiconductor physics! Surface and interface electronic structure: bands and charge carriers! X-STM image simulation! Defect structure and (formation) energies! Optical properties (light–matter interaction)

Applications with DFT Small molecules and atoms! H2

+, He, H–, H2, Ps2, H3+ H2+H+ H2

++H, so far! Finite temperature included! Exact correlation, nodal surfaces for exchange

Continuous/periodic systems! Two-component Fermi sea: e– p+, e– e+

! LDA type correlation functional for the above

Ab initio chemical reactions! Dissociation–recombination in thermal equilibr.! Model for on or beyond BO-PES! Work for time dependence in progress! Fundamental issues in classical to quantum transition: decoherence

T>0 Ab Initio Quantum Chemistry


ELECTRONIC STRUCTURE THEORY

http://www.tut.fi/semiphys

22

Thanks to all of the people involved,• Ilkka Kylänpää and• Markku Leino,in particular.

4.1. Fermion and boson density matrices

Table 4.2: Total energies for He triplet, Li and Be calculated using the free particle(FP) and the Hartree–Fock (HF) nodes with ! = 0.03E!1

h and M = 216, i.e. T !160 K. Also, the best matches from Table 4.1 and accurate reference energies withneeded decimals are given. Energies are given in units of hartree.

FP nodes ”Level nodes” HF nodes exactHe (S = 1) "2.116(2) "2.1579(8) "2.1757(9) "2.1752 a

Li "7.400(6) "7.472(3) "7.4715(30) "7.4781 b

Be "14.596(12) "14.70(3) "14.6845(50) "14.6674 c

a Bürgers et al. [1995]b McKenzie and Drake [1991]c Stanke et al. [2007]

Figure 4.1: Simulation snapshots. Black, blue, red and green refer to electrons, andmagenta is for the positron. Li nuclei cannot be seen in the figures. (LEFT) Fullynonadiabatic LiH molecule simulation giving total energy of "8.0094(32)Eh . Thisis not in good agreement with "8.0661557638Eh by Scheu et al. [2001]. How-ever, the binding energy of about 0.1Eh is relatively close to the accurate value ofabout 0.088Eh. (RIGHT) e+Li molecule with fixed Li nucleus giving total energyof "7.5324(26)Eh . This coincides exactly with "7.5323955Eh calculated using theexplicitly correlated gaussians [Mitroy, 2004].

53


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,

! = 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


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,

! = 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 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

(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).

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).



COMPOSITIONS

26

5.4. H+3 molecular ion

!1.3 !1.2 !1.1 !1.00

5

10

15

20

25

30 H3+ H

2+H+ H

2++H 2H+H+BTL

E (units of hartree)

Tota

l ene

rgy

dist

ribut

ion

Figure 5.8: Histogram of total energy sampling pinned in boxes of width 0.001EH

from (2 ! 104) ! 105, or more, Monte Carlo samples averaged over blocks of 105

samples. The energy expectation values are also given with 2SEM error estimates.The temperature and Trotter number are " 5139.6 K and 2048, respectively. Thehistograms are normalized to unity for all three densities. Other notations are takenfrom Fig. 5.7.

three ranges, i.e. about 0 # 4000 K, about 4000 # 10000 K and above 10000 K, aresubject to changes with larger variation of densities.Above 10000 K in our lowest density case the thermal ionization of H atoms is

evident, see Fig. 5.7, but for our higher density cases some 15000K is needed to bringup first signs of ionization. Similar trend for the ionization is stated in Koskinen et al.[2010], although there the density is notably less than our lowest one.Let us now consider the dissociation–recombination reaction chain, Eq. (5.21),

and the contributing fragments to the quantum statistical NV T equilibrium trying togive an intuitive classical-like picture of the composition. At zero Kelvin ! would beinfinite, however, at finite T we have finite !, that brings classical nature to the systemthe more, the higher the temperature. In other words, the partial decoherence in ourfive particle quantum system increases with increasing temperature, that enables usto distinguish the fragments as separate molecules and atoms in thermal equilibrium.Based on this interpretation, we show the total energy distributions in Fig. 5.8 from

68

The equilibrium compositions of fragments at about 5000 K for the different densities.


MORE COMPOSITIONS

Text

27

5.4. H+3 molecular ion

!1.3 !1.2 !1.1 !1.00

5

10

15

20

25

30 H3+ H

2+H+ H

2++H 2H+H+BTL

E (units of hartree)

Tota

l ene

rgy

dist

ribut

ion

!1.3 !1.2 !1.1 !1.00

5

10

15

20

25

30 H3+ H

2+H+ H

2++H 2H+H+BTL

E (units of hartree)To

tal e

nerg

y di

strib

utio

n

Figure 5.9: Simulation total energy distributions. (LEFT) T ! 4498.2 K, (RIGHT)T ! 6070.3 K. Notations are taken from Fig. 5.8.

sampling the imaginary time paths at about 5000KwithM = 2048 for all considereddensities.For example, for our highest density (gray in Fig. 5.8) we see three main peaks,

and by inspection of that energy distribution the first and the second can clearly beassigned to the rovibrationally excited H+

3 and H2+H+, respectively. As there areno rovibrational excitations available for 2H+H+, the average position of the thirdmain peak is very close to "1Eh. The fourth fragment, H+

2 +H, can be identifiedas the small high-energy side shoulder of H2+H+ peak. With the interpretation ofthe area under the peak as the abundance of the fragment in the equilibrium we findthis contribution to be much smaller than that of the others. In Fig. 5.9 the sameis illustrated for two different temperatures, about 4500 K (left) and 6070 K (right),also. There at the higher temperature the amount of H+

2 is found increased accordingto the same interpretation as above.It is important to note, however, that the above illustration is dependent on the

block averaging procedure, see the caption of Fig. 5.8. Pinning the energy data ofeach and every sample, i.e. choosing block of size one sample, would broaden thepeaks in Fig. 5.8. At the opposite limit, all samples in one block, would give thesingle mean energy or the ensemble average corresponding to the quantum statisticalexpectation value. From the highest density to the lowest the expectation valuesare "1.169(29)Eh , "1.020(33)Eh and "0.9995(4), respectively, Figs. 5.7 and 5.8,where the statistical uncertainty decreases with increasing simulation length.

69


H2+ MOLECULE, REVISITED

Coupled quantum dynamics of the electron and nuclei (of the three-particle system)

• Compare• adiabatic dynamics of e (in BO) and nuclei (ro-vibr)• three-particle dynamics (”all-quantum”, AQ)

• Isotope effect

e–

p+ p+

28


H2+ AND D2

+ MOLECULES: ISOTOPE EFFECT

Adiabatic nucleardynamics: D2

+

-> isotope effect H2+

Distribution widths:0.45 a0 and 0.54 a0,

respectively

M = 26

2.007 a0

2.019 a0

29


THREE-BODY H2+ MOLECULE

Adiabatic nuclear dynamics ”All-quantum” (AQ) dyn.• Widths: 0.54 and 0.58 a0.

• Bond length increase:

anharm. centrifug.

• ET = -0.5987 more accurate

than the adiabatic one.• Zero-point E:

2.075 a0

2.019 a0

30


H2+ MOLECULE: DISTRIBUTIONS

Electron–nucleuspair distribution function

Snapshot from AQ

H atom

R = 2.0 a0

AQ

H atom+ proton

31


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+

Ceperley (95),Storer (68).

32


PS2 @ 0K – 900K

We find:• EPs = – 0.250• EPs2 = – 0.5154• ED = 0.0154 ≈ 0.435 eV :≈ 5000 K

Ps2 @ T ≤ 900 K

Ps

Ps2 @ T = 0 K

T ≤ 900 K

33


PS2 EXPERIMENTAL OBSERVATION ?

History:• D.B. Cassidy & A.P. Mills Jr.: Observation of Ps2 molecule

formation in porous silica [Nature 449, 195 (2007), PRL 100, 013401 (2008)]:

• Below 800–900 K, only; activation energy ~0.074 eV• Rule out the dissociation, because expected at ~0.4 eV• Activation energy 0.074 eV due to Ps2 desorption

• Our observation of surprising Ps2 molecule instability above about 900K in PIMC simulation: ~0.082 eV

• We predict dissociation at about 1000 K, which remains to be experimentally observed!

34


PS2 FREE ENERGY

Free energy is T dependent:

• Ps2 minimizes F = U – TS by being mostly apart above a certain temperature T.

• F depends on entropy S, too, i.e. on Ps density ρ T => Find ρ dependence: Ps2 Ps + Ps ρ2

35

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