elements of molecular simulation methods

Elements of Molecular Simulation Methods

Lecture Note for the Postgraduate Course

Advanced Quantum Theory of Chemical Processes

Koji Ando

School of Chemistry, University of Birmingham, UK

Last updated : January 5, 2004

Contents

1 Potential Energy Concept 51.1 Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Adiabatic Approximation (Born-Oppenheimer) . . . . . . . . . . . . 61.3 Non-adiabatic couplings . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Mixed Quantum-classical Simulation . . . . . . . . . . . . . . . . . . 81.5 Adiabatic and Diabatic Representations . . . . . . . . . . . . . . . .91.6 Three Approximations . . . . . . . . . . . . . . . . . . . . . . . . .10

1.6.1 Crude Adiabatic Approximation . . . . . . . . . . . . . . . .111.7 Some properties of the non-adiabatic couplings . . . . . . . . . . . .11

2 Electronic Structure Theories 132.1 Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . .132.2 Electron Correlation Problem . . . . . . . . . . . . . . . . . . . . . .16

2.2.1 Static and Dynamic Correlations . . . . . . . . . . . . . . . .162.2.2 Various Methods . . . . . . . . . . . . . . . . . . . . . . . .16

2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . .162.3.1 Kohn-Sham theory . . . . . . . . . . . . . . . . . . . . . . .162.3.2 Hybrid Hartree-Fock / DFT . . . . . . . . . . . . . . . . . . 16

2.4 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

3 Molecular Dynamics Simulation 173.1 Summary of Classical Mechanics . . . . . . . . . . . . . . . . . . . .17

3.1.1 Newtonian EOM . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 The Variational Principle . . . . . . . . . . . . . . . . . . . .173.1.3 Hamilton Formalisms . . . . . . . . . . . . . . . . . . . . . .17

3.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1 Verlet algorithm . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Leap-frog algorithm . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Velocity-Verlet algorithm . . . . . . . . . . . . . . . . . . . . 19

3.3 Treatment of Molecules . . . . . . . . . . . . . . . . . . . . . . . . .193.3.1 Multiple time step method (r-RESPA) . . . . . . . . . . . . .20

3.4 Constant Temperature and Pressure Methods . . . . . . . . . . . . . .213.4.1 Common Statistical Ensembles . . . . . . . . . . . . . . . .213.4.2 Temperature and Pressure from MD . . . . . . . . . . . . . .22

3

4 CONTENTS

3.4.3 Constant NVT MD . . . . . . . . . . . . . . . . . . . . . . . 223.4.4 Constant Pressure MD . . . . . . . . . . . . . . . . . . . . .23

4 Adiabatic Molecular Dynamics 254.1 Empirical Force-Field . . . . . . . . . . . . . . . . . . . . . . . . . .264.2 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . .264.2.2 TDSCF MD . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.3 Born-Oppenheimer MD . . . . . . . . . . . . . . . . . . . .264.2.4 Hartree-Fock BO MD . . . . . . . . . . . . . . . . . . . . . 274.2.5 Car-Parrinello MD . . . . . . . . . . . . . . . . . . . . . . . 28

5 Monte Carlo Simulation 295.1 Standard MC (Metropolis algorithm) . . . . . . . . . . . . . . . . . .295.2 Umbrella Sampling Technique . . . . . . . . . . . . . . . . . . . . .30

6 Data Analysis 336.1 Static Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .336.2 Radial disribution function . . . . . . . . . . . . . . . . . . . . . . .346.3 Time correlation function . . . . . . . . . . . . . . . . . . . . . . . .34

6.3.1 Relaxation Phenomena . . . . . . . . . . . . . . . . . . . . .356.3.2 Various Applications . . . . . . . . . . . . . . . . . . . . . . 36

6.4 Free Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .36

7 Time-dependent Quantum Theory 397.1 Perturbative expansion . . . . . . . . . . . . . . . . . . . . . . . . .407.2 First-order perturbation . . . . . . . . . . . . . . . . . . . . . . . . .417.3 Second-order perturbation . . . . . . . . . . . . . . . . . . . . . . .417.4 Schrodinger vs Interaction Pictures . . . . . . . . . . . . . . . . . . .427.5 Interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . .427.6 Time-ordered Exponential . . . . . . . . . . . . . . . . . . . . . . .437.7 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . .43

7.7.1 State-to-state form . . . . . . . . . . . . . . . . . . . . . . .447.7.2 Periodic interaction . . . . . . . . . . . . . . . . . . . . . . .44

8 Light-Matter Interaction 458.1 Electromagnetic wave . . . . . . . . . . . . . . . . . . . . . . . . . .45

8.1.1 “Minimal” Electromagnetic Interaction . . . . . . . . . . . . 458.1.2 Absorption and Emission Spectra . . . . . . . . . . . . . . .45

8.2 Dipole Approximation (long wavelength) . . . . . . . . . . . . . . .468.2.1 Simpler derivation ofUkm . . . . . . . . . . . . . . . . . . . 46

8.3 Density of states of photon field . . . . . . . . . . . . . . . . . . . .478.3.1 Calculation ofρphoton(hω) . . . . . . . . . . . . . . . . . . . 478.3.2 Natural lifetime of excited states . . . . . . . . . . . . . . . .488.3.3 Rate of photo-absorption . . . . . . . . . . . . . . . . . . . .48

8.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48

CONTENTS 5

8.4.1 Einstein’s A and B factors . . . . . . . . . . . . . . . . . . .488.4.2 Oscillator strength . . . . . . . . . . . . . . . . . . . . . . .488.4.3 Correspondence with experiments . . . . . . . . . . . . . . .498.4.4 1st correction to long-wavelength approx. . . . . . . . . . . .49

8.5 Franck-Condon Factor . . . . . . . . . . . . . . . . . . . . . . . . .50

9 Time Correlation Functions 519.0.1 Time-dependent form . . . . . . . . . . . . . . . . . . . . . .529.0.2 Gaussian Wavepacket . . . . . . . . . . . . . . . . . . . . . .52

9.1 Cumulant expansion method . . . . . . . . . . . . . . . . . . . . . .539.1.1 Condon Approximation : . . . . . . . . . . . . . . . . . . . .539.1.2 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.1.3 Cumulant expansion . . . . . . . . . . . . . . . . . . . . . .549.1.4 Cumulant expansion of time-ordered exponential . . . . . . .55

9.2 Time correlation functions . . . . . . . . . . . . . . . . . . . . . . .559.2.1 Ergodic hypothesis . . . . . . . . . . . . . . . . . . . . . . .56

9.3 TCF and spectral line shape . . . . . . . . . . . . . . . . . . . . . . .569.3.1 Exponential TCF . . . . . . . . . . . . . . . . . . . . . . . .569.3.2 Gaussian TCF . . . . . . . . . . . . . . . . . . . . . . . . .569.3.3 Damped-oscillating TCF . . . . . . . . . . . . . . . . . . . .569.3.4 Example of Gaussian TCF . . . . . . . . . . . . . . . . . . .569.3.5 Example of Exponential TCF . . . . . . . . . . . . . . . . .579.3.6 Example : Brownian oscillator model . . . . . . . . . . . . .57

9.4 Motional narrowing . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.4.1 “motional narrowing” . . . . . . . . . . . . . . . . . . . . . 589.4.2 Physical interpretation . . . . . . . . . . . . . . . . . . . . .58

10 Dynamics in Condensed Phases 5910.1 Phenomenology of Brownian Motions . . . . . . . . . . . . . . . . .59

10.1.1 Coarse graining . . . . . . . . . . . . . . . . . . . . . . . . .5910.1.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . .59

10.2 Microscopic model for GLE . . . . . . . . . . . . . . . . . . . . . . 6010.2.1 System + Harmonic bath . . . . . . . . . . . . . . . . . . . .60

10.3 Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . . . .6110.4 Matrix partitioning method . . . . . . . . . . . . . . . . . . . . . . .61

10.4.1 Matrix partitioning . . . . . . . . . . . . . . . . . . . . . . . 6210.5 Projection operator methods (1) . . . . . . . . . . . . . . . . . . . .6210.6 Projection operator methods (2) . . . . . . . . . . . . . . . . . . . .6210.7 Phase Space Distribution Function . . . . . . . . . . . . . . . . . . .63

11 Theory of Liquids 65

6 CONTENTS

12 Reaction Rate Theory 6712.1 Definition in phase space (Classical mechanics) . . . . . . . . . . . .6712.2 Microcanonical ratek(E) . . . . . . . . . . . . . . . . . . . . . . . . 6712.3 Canonical ratek(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.3.1 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.4 Transition state theory (TST) . . . . . . . . . . . . . . . . . . . . . .68

12.4.1 MicrocanonicalkTST(E) . . . . . . . . . . . . . . . . . . . 6912.4.2 CanonicalkTST(T ) . . . . . . . . . . . . . . . . . . . . . . . 69

12.5 Quantum mechanical rate constant . . . . . . . . . . . . . . . . . . .7012.6 Correction from TST . . . . . . . . . . . . . . . . . . . . . . . . . .71

12.6.1 Grote-Hynes theory . . . . . . . . . . . . . . . . . . . . . . .7112.6.2 Kramers limit . . . . . . . . . . . . . . . . . . . . . . . . . . 72

12.7 κ(t) from Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.7.2 An alternative view . . . . . . . . . . . . . . . . . . . . . . . 75

13 Quantum Simulations 7713.1 Wave Packet Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .7713.2 Gaussian Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . .7713.3 Time-dependent Variational Principle . . . . . . . . . . . . . . . . .7713.4 Semiquantal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .7713.5 Semiquantal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .77

13.5.1 1-Dim case . . . . . . . . . . . . . . . . . . . . . . . . . . .7713.5.2 2-Dim case . . . . . . . . . . . . . . . . . . . . . . . . . . .78

13.6 Path-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

14 Quantum Liouville Equation 7914.1 Pure and Mixed States . . . . . . . . . . . . . . . . . . . . . . . . .79

14.1.1 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . 8014.1.2 Example : 2-level system . . . . . . . . . . . . . . . . . . . .8014.1.3 Liouville operator . . . . . . . . . . . . . . . . . . . . . . . . 80

14.2 Reduced density operator . . . . . . . . . . . . . . . . . . . . . . . .8114.3 Wigner Transformation . . . . . . . . . . . . . . . . . . . . . . . . .8114.4 Mixed Quantum-classical Simulations . . . . . . . . . . . . . . . . .81

Chapter 1

Potential Energy Concept

We begin with an introduction of the adiabatic approximation together with its limita-tions described by the nonadiabatic couplings. Theconceptof potential energy surface(PES) thus derived is at the center of our understanding and description of molecularsystems. Indeed the PES concept is so convenient and powerful that we often (per-haps unconsciously) ignore its approximate nature, as would be particularly the casein undergraduate teachings of physical chemistry. Nevertheless, there are variety of in-teresting and important molecular processes where the adiabatic approximation breaksdown and the nonadiabatic transitions and vibronic couplings play significant roles.Also given here is an introductory account of the nonadiabatic molecular dynamicssimulation method, which serves to demonstrate the common basic idea of the relatedmethods. More detailed and advanced theories will be expanded in Chapter X.

1.1 Molecular Hamiltonian

The non-relativistic Hamiltonian of molecular systems consists of electronic and nu-clear degrees of freedom,

H = TN + Te + VeN + Vee + VNN (1.1)

whereT ’s are the kinetic energies andV ’s are the Coulomb interaction between thecharged particles. For example, the nuclear kinetic energy is

TN = −∑I

h2

2MI∇2I

and the interaction between the electrons and nuclei is

VeN = −∑I

∑i

ZIe2

|RI − ri|

whereRI andri are the positions of the nuclei and electrons.ZI andMI are the atomicnumber and mass of theIth nucleus.. More explicitly, the molecular Hamiltonian is

7

8 CHAPTER 1. POTENTIAL ENERGY CONCEPT

written as

H = −∑I

h2

2MI∇2I−∑i

h2

2m∇2i−∑I

∑i

ZIe2

|RI − ri|+∑i<j

e2

|ri − rj |+∑I<J

ZIZJe2

|RI −RJ |(1.2)

Ideally, we wish to solve the whole problem by solving the Schrodinger equation,

HΨ(R, r) = EΨ(R, r) (1.3)

But this is mostly too tall an order for systems complex enough to be of chemicalinterest.1 It would be useful to remember the hydrogen atom consisting of an electronand a proton. Separating out the center of mass motion, the problem reduces to theSchrodinger equation for a single degree of freedom having thereduced massof

µ =mMH

m+MH=

m

m/MH + 1

wherem andMH are the masses of the electron and proton, respectively. The Schrodingerequation has the same form as that for an electron interacting with a fixed proton, ex-cept that the bare electronic massm is replaced by the reduced massµ. Consideringthe mass ratio ofm/MH ' 1.8 × 103, we observeµ ' m. This suggests that solv-ing the electronic problem under the fixed nuclei would be a good approximation. Wetherefore define theelectronic Hamiltonian

He = H − TN = Te + VeN + Vee + VNN

simply by removing the nuclear kinetic energy from the full molecular Hamiltonian.

1.2 Adiabatic Approximation (Born-Oppenheimer)

Now the prescription would be composed of the following four stages:

1. Fix the nucleiR and solve the electronic problem

Heϕn(r; R) = Wn(R)ϕn(r; R) (1.4)

whereϕn(r; R) denotes the electronic wavefunctions. As we will repeat thecalculations by varying the nuclear geometry, the electronic wavefunctions areparametrically dependent onR). Wn(R) denotes the electronic energy levelsat the geometryR.

2. Repeat the electronic calculations by varyingR to obtain the potential energysurface (PES)Wn(R)

Diagram : Potential energy curvesW1 andW2 (diatomic)

1The difficulty would be easily imagined from the fact that we still do not have the analytical solution fora three particle systems such as the helium atom. The problem is not limited to quantum mechanics, but alsoseen in classical mechanics of general interacting three-body system

1.3. NON-ADIABATIC COUPLINGS 9

These two stages would correspond to the narrow definition of “quantum chem-istry”.

3. Examine the nuclear dynamics on the PESWn(R). Here we have several op-tions:

• Calculation of the quantum energy levels: We first define the nuclear Hamil-tonian on then-th PES

H(N)n ≡ TN +Wn(R)

and solve the nuclear Schrodinger equation

H(N)n χv(R) = En,vχv(R)

to obtain the nuclear wavefunctionsχv(R) and the energy levelsEn,v. Thesimplest analysis invokes the harmonic approximation around the bottomof the PES to obtain thevibrational normal modes.

• Wave packet simulations of the quantum dynamics:

ih∂

∂tχn(R) = H(N)

n χv(R)

• Classical molecular dynamics (MD) simulations:

MIR = −∂Wn(R)∂R

• Statistical mechanics simulations in which the Boltzmann distribution onthe adiabatic PES proportional to

exp(−Wn(R)/kBT )

is the key. Representative are the Monte Carlo simulations which includequantum and classical versions. (See Chapter X.)

4. Analysis of the simulation results (statistical or dynamical). This will be thetopic of Chapter X.

1.3 Non-adiabatic couplings

We first expand the total wavefunction by the set of electronic wavefunctionsϕn(r; R)

Ψ(r,R) =∑n

χn(R)ϕn(r;R) (1.5)

The Schrodinger equation with the total HamiltonianH = TN +He is then

[TN +He]Ψ(r,R) = EΨ(r,R) (1.6)


Putting (1.5) into (1.6), multiplyingϕ∗k and integrating over the electronic coordinate,we obtain∑

n

〈ϕk(r;R)|TN |ϕn(r;R)〉χn(R) +Wk(R)χk(R) = Eχk(R) (1.7)

where we used (1.4) and the orthonormality〈ϕk|ϕn〉 = δkn. Noting thatTN =−∑I(h

2/2MI)∇2I operates to bothϕ(r;R) andχ(R), we get

∑n

[〈ϕk|TN |ϕn〉 −

∑I

h2

MI〈ϕk|∇I |ϕn〉 · ∇I

]χn(R)

+TNχk(R) +Wk(R)χk(R) = Eχk(R)

(1.8)

Here,TN and∇I within 〈ϕ| · · · |ϕ〉 do not operate to the further right.

Adiabatic approximation: Neglect of the first line of (1.8) gives

[TN +Wk(R)]χk(R) = Eχk(R) (1.9)

This has a form of the Schrodinger equation for the nuclear wavefunctionχ(R) on a(single adiabatic) PESWk(R) . The neglected terms are calledNon-adiabatic (NA)couplings,

• 1st order NA coupling :h2

MI〈ϕk|∇I |ϕn〉

• 2nd order NA coupling :h2

2MI〈ϕk|∇2

I |ϕn〉

which will induce mixing and transitions among different electronic states.

1.4 Mixed Quantum-classical Simulation

This method assumes that the nuclear motions follow certaintrajectory R(t) and thewavefunction has a form

Φ(r,R) '∑n

cn(t)ϕn(r; R(t)) (1.10)

The coefficient|cn(t)|2 gives the probability of finding the system in then-th electronicstate. When electronic transitions are involved, it is not a trivial task to define themethod to determine the trajectoryR(t). Most conveniently, it could be the classicaltrajectory on an adiabatic PESWn(R), and then we may switch among the PES’s vianon-adiabatic transitions according to the development of the coefficientscn(t). Thisis called thesurface hoppingtrajectory method.

Diagram : Curve crossing

1.5. ADIABATIC AND DIABATIC REPRESENTATIONS 11

The problem is now to determine the development ofcn(t). To this end we put (1.10)into the time-dependent Schrodinger equation

ih∂Φ∂t

= [TN +He]Φ

which gives

ih∑n

(dcndt

ϕn + cn∂ϕn∂t

) =∑n

cn[TN +Wn(R)]ϕn

As in the previous section, we multiplyϕ∗k and integrate over the electronic coordinates.Noting the orthonormality〈ϕk|ϕn〉 = δkn, we get

ih∂ck∂t

+ ih∑n

cn

⟨ϕk|

∂ϕn∂t

⟩= Wk(R)ck +

∑n

cn〈ϕk|TN |ϕn〉

The last term in the right hand side represents the second-order NA couplings, whichwe will neglect. Then, using the chain rule⟨

ϕk|∂

∂tϕn(r; R(t))

⟩= 〈ϕk|∇R|ϕn〉 · R(t) ≡ dkn(R) · R(t)

we find the first-order NA coupling which has been denoted by the vectordkn(R).This finally gives a set of coupled equations for the probability amplitude coefficientscn(t))

ihd

dtck(t) = Wk(R)ck(t)− ih

∑n

dkn(R) · R(t)cn(t)

1.5 Adiabatic and Diabatic Representations

We will repeat basically the same procedure as in section 1.3 assuming the wavefunc-tion of the form

Ψ(r,R) =∑n

χn(R)ϕn(r;R)

but here we donot require the electronic basisϕn(r;R) to be the eigenfunctions of theelectronic Schrodinger equation (1.4). The electronic Hamiltonian is hence representedby a matrixWnm(R) ≡ 〈ϕnHelϕm. On the other hand, when the electronic basisϕnare the eigenfunctions of the electronic Schrodinger equation which diagonalize thematrix,Wnm(R) = δnmEn(R), they are called as theadiabatic electronic basis.

By following the analogous procedure to that in section (1.3), the Schrodinger equa-tionHΨ = EΨ is converted to

[TN+Wkk(R)+Lkk(R)]χk(R)+∑n 6=k

[Wkn(R)+Lkn(R)]χn(R) = Eχk(R) (1.11)

where we have defined the non-adiabatic coupling operator

Lkn ≡ 〈ϕk|TN |ϕn〉 −∑I

(h2/MI)〈ϕk|∇I |ϕn〉 · ∇I


If we were able to choose the electronic basisϕn(r;R) to minimize the non-adiabaticcouplings〈ϕk|∇I |ϕn〉, we may neglect the terms fromLkn to obtain

[TN +Wkk(R)]χk(R) +∑n 6=k

Wkn(R)χn(R) = Eχk(R) (1.12)

This somehow resembles the previous equation (1.8), but here the mixing and transi-tions among electronic states are induced by the off-diagonal elementsWkn rather thanthe non-adiabatic couplings.

1.6 Three Approximations

The adiabatic approximation, whose basics have been discussed in section 1.2, issometimes classified into three versions: Born-Huang, Born-Oppenheimer and crude-adiabatic approximations.

1. Born-Huang (BH) approximation is obtained from (1.11) after the following as-sumptions:

• The adiabatic electronic basis is employed forϕn, and henceW is rep-resented by the diagonal adiabatic PES,Wkn(R) = δknWn(R).

• Neglect the off-diagonal non-adiabatic coupling operatorsLkn(R)

• Retain the diagonalsLkk(R)

The retainment of the diagonalsLkk(R) is the key characteristic of the BH ap-proximation.

2. Born-Oppenheimer (BO) approximation is obtained by neglecting the diagonalsLkk(R) on top of the Born-Huang approximation. Namely,

• BO = Born-Huang AND the neglect of diagonalsLkk(R)

3. Crude-adiabatic (CA) approximation may be summarized as follows:

• Neglect theR-dependence of the electronic basis by employing the func-tionϕ at particular nuclear geometryR0

Ψ(r,R) '∑n

χn(R)ϕn(r;R0) (1.13)

R0 is usually the equilibrium geometry in the ground state.

• Neglect all the off-diagonal couplings

The CA approximation will be further expanded below.

1.7. SOME PROPERTIES OF THE NON-ADIABATIC COUPLINGS 13

1.6.1 Crude Adiabatic Approximation

As noted above, the CA approximation employs the electronic basisϕ at a particularnuclear geometryR0 from the electronic Schrodinger equation

He(r;R0)ϕn(r;R0) = Wn(R0)ϕn(r;R0)

The electronic Hamiltonian is then expanded around this reference geometry such thattheR-dependence is represented by∆U :

He(r;R) = He(r;R0) + ∆U(r,R)

The use of (1.13) meansLkn = 0 sinceR0 is constant. Hence (1.11) reduces to

[TN +Wk(R0) + ∆Ukk(R)]χk(R) +∑n 6=k

∆Ukn(R)χn(R) = Eχk(R)

where∆Ukn(R) =∫drϕ∗k(r;R0)∆U(r,R)ϕn(r;R0)

In the Crude-Adiabatic approximation, the off-diagonals are further neglected togive finally

[TN +Wk(R0) + ∆Ukk(R)]χk(R) = Eχk(R)

1.7 Some properties of the non-adiabatic couplings

1. The real part of the diagonal first-order NA coupling vanishes:

<〈ϕn|∇I |ϕn〉 = 0

Proof: Take the derivative∇I of 〈ϕn|ϕn〉 = 1, which gives

〈∇Iϕn|ϕn〉+ 〈ϕn|∇Iϕn〉 = 2<〈ϕn|∇I |ϕn〉 = 0

Corollary: If ϕn is a real function,

〈ϕn|∇I |ϕn〉 = 0,

and henceLnn = 〈ϕn|TN |ϕn〉.

2. For adiabatic electronic basisϕn,

(Wn −Wk)〈ϕk|∇R|ϕn〉 = 〈ϕk|∇RV |ϕn〉 (1.14)

Chapter 2

Electronic Structure Theories

According to the PES concept discussed in Chapter 1, the nuclear dynamics evolve onthe PES determined by the electronic HamiltonianHe. Therefore, if we are to pro-ceed beyond the conventional molecular simulations that employ empirical potentialfunctions and force-fields (Chapter X), we inevitablly need to learn molecular elec-tronic structure theories. As it is a huge and nearly mature field, greater number ofexcellent textbooks and review articles are available compared to the field of molec-ular dynamics simulation. In this chapter we survey the minimum basics of the elec-tronic structure theories, including the Hartree-Fock, density functional, and valencebond methods. We will not go into the details of the electron correlation problemsand related computational methods, which should have included the configuration-interaction, multi-configuration self-consistent-field, coupled-cluster, and many-bodyperturbation theories, but will only comment on the future directions and refer to someessential readings.

2.1 Hartree-Fock Method

Electronic Hamiltonian

He = Te + VeN + Vee + VNN

=∑i

(− h2

2m∇2i −

∑I

ZIRIi

)+∑i<j

1rij

+∑I<J

ZIZJRIJ

=∑i

Hcore(i) + Vee + VNN

(2.1)

VNN is just a constant in the electronic problem (adiabatic approximation), so we here-after considerhe ≡ He − VNNTwo-electron molecules (e.g., H2)

he = Hcore(1) +Hcore(2) +1r12

15

16 CHAPTER 2. ELECTRONIC STRUCTURE THEORIES

Slater determinant :ψ(1, 2) = 1√2

∣∣∣∣ ϕ1(1) ϕ1(2)ϕ2(1) ϕ2(2)

∣∣∣∣ = 1√2(ϕ1(1)ϕ2(2)−ϕ2(1)ϕ1(2)) =

1√2|ϕ1ϕ2|

Spin-orbitals :ϕ1(1) = φ1(1)α(1) = φ1(1)

ϕ2(1) = φ1(1)β(1) = φ1(1)(space orbital)× (spin functionα,

β)

Diagram : orbital diagramsφ1φ1, φ1φ2, φ1φ2

Slater-determinants satisfy thePauli-Principle of many-electron systems:

• Anti-symmetry :ψ(2, 1) = −ψ(1, 2)

• Exclusion principle :ψ(1, 2) = 0 if ϕ1 = ϕ2 (space and spin)

Energy : E = 〈ψ(1, 2)|he|ψ(1, 2)〉 =∫ ∫

ψ(1, 2)∗heψ(1, 2)dτ1dτ2

=12

(〈φ1(1)φ1(2)|he|φ1(1)φ1(2)〉−〈φ1φ1|he|φ1φ1〉−〈φ1φ1|he|φ1φ1〉+〈φ1φ1|he|φ1φ1〉)

• The 1st term =〈φ1(1)φ1(2)|Hcore(1) +Hcore(2) + 1r12|φ1(1)φ1(2)〉

= 〈φ1(1)|Hcore(1)|φ1(1)〉〈φ1(2)|φ1(2)〉+ 〈φ1(1)|φ1(1)〉〈φ1(2)|Hcore(2)|φ1(2)〉

+〈φ1φ1| 1r12|φ1φ1〉

≡ 2Hcore11 + J11

• The 4th term gives the same result :2Hcore11 + J11

• The 2nd term =〈φ1|Hcore|φ1〉〈φ1|φ1〉+〈φ1|φ1〉〈φ1|Hcore|φ1〉+〈φ1φ1| 1r12|φ1φ1〉

= 0 because〈α|β〉 = 0

• The 3rd term is also zero.

Thus,E = 2Hcore11 + J11

Let us next consider an open-shell singlet configuration|φ1φ2| (α electron inφ1 andβin φ2)In the similar way, we get

E = Hcore11 +Hcore

22 + J12

whereJ12 ≡ 〈φ1φ2| 1r12|φ1φ2〉 ≡ 〈12||12〉 = J21 = 〈φ2φ1| 1

r12|φ2φ1〉 ≡ 〈21||21〉

Triplet configuration|φ1φ2| (α electrons in bothφ1 andφ2) gives

E = Hcore11 +Hcore

22 + J12 −K12

2.1. HARTREE-FOCK METHOD 17

whereK12 ≡ 〈12||21〉 = K21

Jij andKij are calledCoulomb andExchangeintegrals, respectively.

These examples suggest the following rule to write downenergiesof electron config-urations

• Each electron in orbitalφi contributesHcoreii

• Each electron pair in orbitalsφi andφj contributesJij (regardless of the spin)

• Each electron pair of the same spinin orbitalsφi andφj contributes−Kij

Exercise : Write down the energy expression for the following electron configurations.

1. α andβ electron pair inφ1 and anα electron inφ2 (open-shell doublet)

2. paired (α andβ) electrons in bothφ1 andφ2 (closed-shell)

Answer :

1. E = 2Hcore11 +Hcore

22 + J11 + 2J12 −K12

2. E = 2Hcore11 + 2Hcore

22 + J11 + J22 + 4J12 − 2K12

In general, for closed-shellN -electron systems,

E =N/2∑i=1

Hcoreii +

N/2∑i=1

Jii +N/2∑i=1

N/2∑j=i+1

(4Jij − 2Kij)

This can be simplified by notingJii = Kii

E =N/2∑i=1

Hcoreii +

N/2∑i=1

N/2∑j=1

(2Jij −Kij)

Hartree-Fock Method= find variationally bestMOsφi under the orthonormality condition〈φi|φj〉 = δij⇒ minimizeL ≡ E +

∑i

∑j εij(δij − 〈φi|φj〉) w.r.t. the variation of MOsφi →

φi + δφi(whereεij is the Lagrange multiplier)⇒ Hartree-Fock equation :Fiφi =

∑j εijφj

where the Fock-operator is defined asFi ≡ Hcore +N/2∑j=1

(2Jj−Kj) [for closed-shell

systems]The Coulomb and Exchange operatorsJj andKj are defined by

Jj(1)φi(1) = 〈φj(2)| 1r12|φj(2)〉φi(1)

Kj(1)φi(1) = 〈φj(2)| 1r12|φi(2)〉φj(1)

18 CHAPTER 2. ELECTRONIC STRUCTURE THEORIES

Comment : The bottleneck for the first-learner would be the abstractness of the func-tional variation of MOsφi → φi + δφi. In practical calculations, however, we employLCAO-MO expansion of the MOsφi =

∑ν

cνiχν whereχ is the atomic orbitals

(AO), and optimize the coefficientscνi. This reduces the problem to a (non-linear)matrix eigenvalue problem which is much more handy for computer implementations.(Hartree-Fock-Roothaan-Hall method)

Summary of Hartree-Fock method

• Assumes a single Slater-determinant for the electronic wavefunction,which satisfies the Pauli Principle of many-electron systems

• and variationally optimizes the MOs by minimizing the exact en-ergy expression for the Slater-determinant wavefunction under theorthonormality condition of the MOs.

• The electronic energy is expressed by one-electron integralsHcoreii

and Coulomb and Exchange two-electron integralsJij andKij .

2.2 Electron Correlation Problem

2.2.1 Static and Dynamic Correlations

2.2.2 Various Methods

• CI, MPn, MCSCF, MRCI, CC, MRMP, etc.

2.3 Density Functional Theory

2.3.1 Kohn-Sham theory

2.3.2 Hybrid Hartree-Fock / DFT

2.4 Other Methods

• Valence-Bond Method

– non-orthogonal orbitals

– chemically intuitive resonance structures

• Semi-empirical MO Methods

– Approximations: neglect of differential overlaps, empirical parameters

– CNDO, INDO, MINDO, PM3, AM1, etc. etc.

Chapter 3

Molecular Dynamics Simulation

3.1 Summary of Classical Mechanics

3.1.1 Newtonian EOM

mx = F = −∂V (x)∂x

3.1.2 The Variational Principle

Action : I ≡∫ t2

t1

L(x, x)dt

The classical trajectoryx(t1) → x(t2) minimizes the actionI against small variationδx(t) (with fixed endsδx(t1) = δx(t2) = 0).

δI =∫ t2

t1

dt δL =∫ t2

t1

dt

(∂L∂x

δx+∂L∂x

δx

)

=∫ t2

t1

dt

∂L∂x− d

dt

(∂L∂x

)δx+

∂L∂x

δx

∣∣∣∣t2t1

Stationary conditionδI = 0 for arbitraryδx(t)

⇒ Euler-Lagrange eq :∂L∂x− d

dt

(∂L∂x

)= 0

This is easily seen to give the Newtonian EOM forL = T − V =m

2x2 − V (x)

3.1.3 Hamilton Formalisms

Hamiltonian EOM : Momentum and Hamiltonian are defined by

p ≡ ∂L∂x

and H ≡ px− L

19

20 CHAPTER 3. MOLECULAR DYNAMICS SIMULATION

For Cartesian coordinates, we getp = mx andH = p2/2m + V (x), and it is easy tosee that the following Hamiltonian EOM is equivalent to Newtonian EOM:

x =∂H

∂pand p = −∂H

∂x

Lagrange and Hamilton theories are more flexible and convenient when dealing withgeneral coordinate systems other than the Cartesian.

3.2 Integration

3.2.1 Verlet algorithm

r(t+ δt) = r(t) + r(t)δt+ 12 r(t)δt2 + · · ·

= r(t) + v(t)δt+ 12a(t)δt2 + · · ·

r(t− δt) =

⇒ r(t+ δt) = 2r(t)− r(t− δt) + a(t)δt2

• error is of orderO(δt4)

• time-reversible

• requires storage of the previous positionr(t+ δt)

• small terma(t)δt2 is added to a difference of large terms2r(t)− r(t− δt)⇒ numerical round-off imprecisions

• velocities are unnecessary to evolve the trajectory, but needed when calculatingthe kinetic energy

v(t) =r(t+ δt)− r(t− δt)

2δt

– error is of orderO(δt3)

– small difference is divided by the small timestep⇒ numerical imprecisions

3.2.2 Leap-frog algorithm

Propagate position and velocity

r(t+ δt) = r(t) + v(t+ δt2 )δt

v(t+ δt2 ) = v(t− δt

2 ) + a(t)δt

3.3. TREATMENT OF MOLECULES 21

• mathematically equivalent to Verlet method (easily verified by eliminating thevelocities)

• velocities (& kinetic energy) at timet

v(t) =v(t+ δt

2 ) + v(t− δt2 )

2

• advantages over the original Verlet

– less problematic on the numerical round-off due to taking differences

– explicit appearance of velocities

– criticisms : treatment of velocities — still not very satisfactory

3.2.3 Velocity-Verlet algorithm

r(t+ δt) = r(t) + v(t)δt+ 12a(t)δt2

v(t+ δt2 ) = v(t) + a(t) δt2

v(t+ δt) = v(t+ δt2 ) + a(t+ δt) δt2

• mathematically equivalent to the previous two

3.3 Treatment of Molecules

• Time scales :bond stretch< bend< torsion< collective motions< rotation< translation

• Fixing bond lengths (to save CPU)

1. treat as a rigid body (small molecules such as H2O)

2. transform the EOM to internal coordinates (e.g., GF-matrix method)

3. introduce bond constraint condition to the Lagrangian⇒ constrained EOM (SHAKE and RATTLE methods)

• Multiple time step method(s)

– small time step for fast motions

– frequent update of short-range interactions


3.3.1 Multiple time step method (r-RESPA)

r-RESPA = reversible REference System Propagator AlgorithmClassical Liouville operator :

iL ≡ ...,H =f∑i=1

[∂H

∂pi

∂

∂qi− ∂H

∂qi

∂

∂pi

]

Poisson bracket :A,B =f∑i=1

[∂A

∂qi

∂B

∂pi− ∂A

∂pi

∂B

∂qi

]Propagation of the phase-space pointΓ = qi(t), pi(t)

Γ(t+ ∆t) = eiL∆t Γ(t)

In Cartesian coordinate withH =∑i

p2i

2m i+V (x), pi = mivi,

(Fi = − ∂V

∂xi= force

)

iL =∑i

[vi

∂

∂xi+Fimi

∂

∂vi

]

Usingec∂∂y f(y) = f(y + c), we find :

• ev∆t ∂∂x propagatesx to x+ v∆t

• e∆t Fm∂∂v propagatesv to v + F

m∆t

Trotter decomposition

ei(L1+L2)∆t = eiL1∆t/2eiL2∆teiL1∆t/2 +O(∆t3)

If we choose : iL1 = Fm

∂∂v , iL2 = v ∂

∂x

Γ(t+ ∆t) = e∆t2Fm

∂∂v e∆tv ∂

∂x e∆t2Fm

∂∂v · Γ(t) +O(∆t3)

From right to left :

• e∆t2Fm

∂∂v propagatesv to v + F

m∆t2

• e∆tv ∂∂x propagatesx to x+ v∆t

• e∆t2Fm

∂∂v propagatesv to v + F

m∆t2 (with updated forceF (x) )

This is exactly the Velocity-Verlet algorithm

3.4. CONSTANT TEMPERATURE AND PRESSURE METHODS 23

Decomposition of forces : (fast/slow, tight/soft, short/long-range etc.)

F = Ffast + Fslow

Accordingly, iL1 = Fslowm

∂∂v iL2 = v ∂

∂x + Ffastm

∂∂v ,

Then, the propagator will be

e∆t2

Fslowm

∂∂v e∆t(v ∂

∂x+Ffastm

∂∂v

)e∆t2

Fslowm

∂∂v

We further decompose the propagator in the middle inton micro-stepswith δt ≡ ∆t/n

e∆t2

Fslowm

∂∂v

[eδt2

Ffastm

∂∂v eδtv

∂∂x e

δt2

Ffastm

∂∂v

]ne

∆t2

Fslowm

∂∂v

Implementation It might be easier to see the code :

δt = ∆t/n ! micro-timestep (for fast motions)do istep = 1, nstep ! overall simulation steps

v = v + (∆t/2) · (Fslow/m)do j = 1,n ! inner loop for fast motions

v = v + (δt/2) · (Ffast/m)x = x + δt · vcall calculateforce(Ffast)v = v + (δt/2) · (Ffast/m)

end docall calculateforce(Fslow)v = v + (∆t/2) · (Fslow/m)

end do

3.4 Constant Temperature and Pressure Methods

3.4.1 Common Statistical Ensembles

• constant NVE (microcanonical)

• constant NVT (canonical)

• constant NPT (isothermal-isobaric)

• constantµVT (grand canonical) [µ = chemical potential]

Straightforward / standard use of :

• MD ⇒ microcanonical (energy conservation of classical mech.)

• MC⇒ canonical (Metropolis algorithm)


Statistical average (constant temperature)

〈A(q, p)〉 =1Q

∫ ∫dp dq A(q, p) e−H(q,p)/kBT

whereQ is the partition function :Q ≡∫ ∫

dp dq e−H(q,p)/kBT

3.4.2 Temperature and Pressure from MD

Equipartition theorem :

〈 p2αi

2mi〉 =

kBT

2(α = x, y, z)

i.e. average kinetic energy ofkBT/2 for each degree of freedom

The kinetic temperature is thus computed in the classical MD by

T =1

3NkB〈∑i

p2i

mi〉

Pressure from MD : PV = NkBT + 〈W 〉

Virial : W ≡ 13

N∑i=1

ri · f inti

(f inti = −∂V

∂ri

)

3.4.3 Constant NVT MD

Extended system method(Nose-Hoover)Couple with external heat bath⇒ friction parameterη (fictious massQ)

ri =pimi

, pi = fi −pηQ

pi

η =pηQ, pη =

N∑i=1

p2i

mi−NfkBT

(Nf = number of (unconstrained) degrees of freedom =3N −Nc)• proven to generate canonical ensemble• Conserved quantity (for coding checks) :

E =∑i

p2i

2mi+ V (r) +

p2η

2Q+NfkBTη

3.4. CONSTANT TEMPERATURE AND PRESSURE METHODS 25

3.4.4 Constant Pressure MD

Constant pressureP in simulation ⇐ V olume control(as a dynamical variable)⇒ coordinate scaling :ri = V 1/3si

extended Lagrangian : (Q = fictiousmass forV )

L =12

∑i

miv2i − V (r) +

12QV 2 − PV

Euler-Lagrange eq : ddt∂L∂q −

∂L∂q = 0 for q = si, V

⇒ si =1

V 1/3

fimi− 2

3Vs V , V =

1Q

(P − P )

P ≡ 13V

(∑i

miv2i +W ) (W = virial)

Chapter 4

Adiabatic Molecular Dynamics

According to the adiabatic approximation discussed in Chapter 1, the nuclear dynamicspropagates on a single adiabatic PES. The PES is determined in turn from the electronicstructure as a function of nuclear configurations. Therefore, the ‘self-consistency’ be-tween the nuclear and electronic degrees of freedom is desired even within the adiabaticapproximation. In this chapter, we overview various levels of treatment, from the mostprimitive ‘pre-determined frozen’ PES to the most advanced ‘on-the-fly evaluation’ ofthe PES.

Functional fitting

• Choice of the functional form (physically adequate asymptotic behavior, sym-metry etc.)

• Empirical parametrization using e.g. spectroscopic data

• Ab initio parametrization using quantum chemical calculations

• Dimensionality problem : when the system hasf degrees of freedom (f = 3N−6 for non-linearN -atoms molecules), and ifM data points are needed per degreeof freedom for the functional fitting, the total number of data points required,Mf , may be prohibitively huge for realistic systems. (e.g.,M ∼ 10 andN = 6requires1012 points.)

On-the-fly evaluation of the potentialWn(R) and gradient∂Wn/∂R

• Ab initio MD, Car-Parrinello MD

• Still computationally expensive, but becoming feasible along with the increaseof the computer power

27

28 CHAPTER 4. ADIABATIC MOLECULAR DYNAMICS

4.1 Empirical Force-Field

Standard FF for MD simulations of biomolecules

W =∑

bonds

KR(R−Re)2 +∑

angles

Kθ(θ − θe)2 +∑

torsions

Vn[1 + cos(nφ− γ)]

+∑

atoms(i<j)

QiQjrij

+∑

atoms(i<j)

4εij

[(σijrij

)12

−(σijrij

)6]

• First three terms = bonding potential

• Last two terms = non-bonding interaction (Electrostatic + Short-range repulsion)

• Inadequate for bond-breaking and -forming processes

• Lack of electronic polarization effects, charge-transfer interaction

4.2 Hybrid Methods

4.2.1 Classical Mechanics

MIRI(t) = − ∂

∂RIWmodele (R(t))

4.2.2 TDSCF MD

• On-the-fly evaluation of the local potentialW (R) = 〈Ψ0(r;R)|He(r;R)|Ψ0(r;R)〉e

• Time-dependent propagation of the (ground state) electronic wavefunctionΨ0(r;R).MIRI(t) = − ∂

∂RI〈Ψ0|He|Ψ0〉

ih∂Ψ0

∂t= HeΨ0

Simple model ofΨ0 is often employed (rather than carrying out quant. chem. calcula-tions), e.g., basis function expansionΨ0(t) =

∑i

ci(t)F (r;R)

4.2.3 Born-Oppenheimer MD MIRI(t) = − ∂

∂RIminΨ0〈Ψ0|He|Ψ0〉

E0Ψ0 = HeΨ0

Optimize the electronic wavefunctionΨ0 at each nuclear configurationR (rather thanpropagating it as in TDSCF MD).

4.2. HYBRID METHODS 29

4.2.4 Hartree-Fock BO MD

If we employ the Hartree-Fock wavefunction :ΨHF0 = detψi

whereψi = HF orbitals (1-electron, orthonormal),

minΨ0〈Ψ0|He|Ψ0〉 ⇒ min

ψi〈ΨHF

0 |He|ΨHF0 〉

∣∣∣∣〈ψi|ψj〉=δij

i.e., minimization within theψi-space under the constraint〈ψi|ψj〉 = δij .

HF Lagrangian : LHFe = 〈ΨHF

0 |He|ΨHF0 〉 −

∑i,j

εij(〈ψi|ψj〉 − δij)

εij = Lagrange multipliersVariational (stationary) condition :

δLHFe

δψ∗i=δLHF

e

δψi= 0 ⇒ Fiψi =

∑j

εij ψj (HF equation)

• Fock-operator :Fi ≡ Hcore +N/2∑j=1

(2Jj − Kj) [for closed-shell systems]

• canonical (diagonal) form :Fiψi = εiψi (εi = orbital energy)

• may also use Kohn-Sham (DFT)FKS and KS orbitals

HF BO MD :MIRI(t) = − ∂

∂RI〈ΨHF

0 |He|ΨHF0 〉 (ΨHF

0 = detψi)

0 = −Fiψi +∑j

εijψj

This set of equations can be derived from an Extended Lagrangian:

LBO =∑I

12MIR

2I − 〈ΨHF

0 |He|ΨHF0 〉+

∑i,j


by assuming that the Euler-Lagrange equationof the classical mechanics applies forboth the nuclear and electronic (orbital) degrees of freedom

⇒ Euler-Lagrange eq :d

dt

∂LBO

∂q− ∂LBO

∂q= 0 for q = RI , ψi, ψ

∗i

(Note : functionalderivatives forψi andψ∗i )

30 CHAPTER 4. ADIABATIC MOLECULAR DYNAMICS

4.2.5 Car-Parrinello MD

Introduce : fictious massand kinetic energyfor electronic (orbital) degrees of freedom⇒ Extended Lagrangian :

LCP =∑I

12MIR

2I +

∑i

12µi|ψi|2 − 〈ΨHF

0 |He|ΨHF0 〉+ constraints

e.g., constraints = MO orthonormality =∑i,j


Euler-Lagrange eq⇒

MIRI(t) = − ∂

∂RI〈ΨHF

0 |He|ΨHF0 〉+

∂

∂RIconstraints

µiψi(t) = − δ

δψ∗i〈ΨHF

0 |He|ΨHF0 〉+

δ

δψ∗iconstraints

Car-Parrinello HF MDMIRI(t) = − ∂

∂RI〈ΨHF

0 |He|ΨHF0 〉 (ΨHF

0 = detψi)

µiψi(t) = −Fiψi +∑j

εijψj

( may also use Kohn-Sham Fock operatorFKS )

• Electronic (orbital) degrees of freedom (El-DoF) are treated as dynamical vari-ables

• No (strict) minimization in the MOψi-space

• Deviate from BO-MD due to thermal fluctuations of the El-DoF

• The dynamics of the El-DoF must be kept cool (eg using constraints)

Diagram : CP-MD trajectory in the coordinate and orbital space

Chapter 5

Monte Carlo Simulation

Monte Carlo integration⇐ random sampling

I =∫ 1

0

f(x)dx ' 1Nsample

Nsample∑i=1

f(xi)

xi = uniform random numbers in [0, 1]

Statistical average (constant temperature)⇐ Phase-space integration

〈A(q, p)〉 =1Q

∫ ∫dp dq A(q, p) e−H(q,p)/kBT

whereQ is the partition function :Q ≡∫ ∫

dp dq e−H(q,p)/kBT

For momentum-independent quantitiesA(q) (with H =∑i p

2i /2mi + V (q))

⇒ Configuration space integration

〈A〉 =1Z

∫dqA(q)e−V (q)/kBT

whereZ =∫dqe−V (q)/kBT

5.1 Standard MC (Metropolis algorithm)

Metropolis MC

• generates configurations R in canonical ensemble

• Core algorithm :

do istep = 1, nstep ! overall Monte Carlo steps· · ·Rnew = Rold + δR ! trial move

31

32 CHAPTER 5. MONTE CARLO SIMULATION

call calculatepotential(V (Rnew))∆V = V (Rnew)− V (Rold)

! Metropolis testif ( ∆V < 0 ) then !Rnew is more stable

acceptRnew

else if (e−∆V/kBT > randomnumber ) then !Rnew is less stable but thermally acceptableacceptRnew

elserejectRnew ! stay atRold

end if· · ·

end do

• Statistical average is calculated by:

〈A〉 =1

Nstep

Nstep∑i=1

A(Ri)

MC vs MD

• direct generation of canonical ensemble (straightforward MD⇒ microcanoni-cal)

• Phase-space / configration-space average (MD analysis⇒ time average assum-ing the ergodic hypothesis)

• no need to evaluate forces

• no time evolution

5.2 Umbrella Sampling Technique

Finite length of MC sampling⇒ the system may be trapped in local potential minima.

In order to extend the sampling to high potential (unstable) configurations, we augmenta bias (weight / window / umbrella) potentialW (q) to the original (unbiased) potentialV (q). The statistical average obtained from this biased simulation is

〈A(q)〉w =1Qw

∫dq A(q) e−β(V (q)+W (q))

(Qw =

∫dq e−β(U(q)+W (q))

)

5.2. UMBRELLA SAMPLING TECHNIQUE 33

The statistics of the original (unbiased) system are reproduced by

〈A(q)〉0 =1Q0

∫dq A(q) e−βV (q) × Qw

Qw

=QwQ0

1Qw

∫dq A(q) e+βW (q)e−β(V (q)+W (q)) =

QwQ0〈A(q) e+βW (q)〉w

=〈A(q) e+βW (q)〉w〈e+βW (q)〉w

= 〈e−βW (q)〉0 〈Ae+βW (q)〉w

34 CHAPTER 5. MONTE CARLO SIMULATION

Chapter 6

Data Analysis

6.1 Static Properties

Ergodic hypothesis : Statistical ensemble average = average over long time

〈A〉 =1τrun

∫ τrun

0

A(t) dt =1

Nrun

Nrun∑k=1

Ak

whereτrun = Nrun∆t (Nrun steps,Ak = A(k∆t))

• RMS deviation =√〈δA2〉 (δA ≡ A− 〈A〉)

Practical

Method 1: save the whole sequence ofA(t)⇒ analyse after simulation

Method 2: ”on-the-fly” summing up.For RMS, we sum upδA = A−〈A〉. However, we don’t know〈A〉 until the endof simulation. The following conversion is then useful:

〈δA2〉 = 〈(A− 〈A〉)2〉 = · · · = 〈A2〉 − 〈A〉2

35

36 CHAPTER 6. DATA ANALYSIS

sum = 0 ; sum2 = 0do istep = 1, nstep ! overall simulation steps

· · ·call calculatequantities(A)sum = sum +Asum2 = sum2 +A2

· · ·end doaverage = sum/nstepvariance = sum2/nstep - average2

RMS =√

variance

6.2 Radial disribution function

Example : Ion pair in water

6.3 Time correlation function

correlation between points with time intervalτ

CAA(τ) = 〈δA(0)δA(τ)〉 =1

trun − τ

∫ trun−τ

0

δA(t) δA(t+ τ) dt

6.3. TIME CORRELATION FUNCTION 37

• For random motions (e.g. in liquids)CAA(t)→ 0 ast→ +∞ (loss of correlation)

• For ”regular” motions (eg. harmonic oscillators / phonons in ”perfect” solids)

〈x(0)x(t)〉 = x(0)2 cosωt

• For (slightly) disordered set of oscillators⇒ “dephasing”

x(t) =∑i

cixi(t) =∑i

cixi(0) cosωt

Example : Dielectric response of water

6.3.1 Relaxation Phenomena

Onsager’s Regression Hypothesis

A(t)− 〈A〉A(0)− 〈A〉

=〈δA(t)δA(0)〉〈δA2〉

non-equil. response equil. correlation

decay of (experimentally) decay of spontaneousprepared non-equil. state fluctuation in equil.

Microscopically proven by Fluctuation-dissipation theorem in the Linear-Responselimit


6.3.2 Various Applications

• Transport properties (Diffusion constant)⇐ velocityTCF

• Microwave & IR spectra⇐ dipoleTCF

• Electronic spectra⇐ transition dipoleTCF

• Electron / excitation transfer rates⇐ TCF of energy gap(Fermi Golden Rule)

• Chemical reaction rates⇐ flux-flux TCF

cf. G C Schatz & M A Ratner, ”Quantum Mechanics in Chemistry” (Prentice Hall,1993)

Diffusion constant ≡mean-squares displacement (Einstein relation)

〈|r(t)− r(0)|2〉 = 6Dt

Relation to velocity TCF :

r(t)− r(0) =∫ t

0

v(τ)dτ ⇒ 6Dt =∫ t

0

dτ1

∫ t

0

dτ2〈v(τ1)v(τ2)〉

∂∂t both sides

6D = 2∫ t

0

dτ〈v(τ)v(t)〉 = 2∫ t

0

dτ〈v(0)v(t− τ)〉

⇐ TCF depends only on time interval(in stationary equilibrium)Changing the integration variable fromτ to τ ′ ≡ t− τ ,

D =13

∫ t

0

dτ ′〈v(0)v(τ ′)〉

6.4 Free Energy Surfaces

Remember that the Gibbs free energy is related to the equilibrium constant and thus theprobability distributionsof the reactant and product species. For example, forA B,

e−∆G/kBT = K =[B][A]

=Prob. BProb. A

This would suggest the following generalization to more general “states” of the system

∆G ≡ G2 −G1 = −kBT ln(

Prob. State 2Prob. State 1

)

6.4. FREE ENERGY SURFACES 39

Now, letX be some coordinate(s) of the system. This may be a position coordinateitself or a function of positions. The free energy curves or surfaces alongX can bedefined and calculated from the probability distribution ofP (X)

G(X) = −kBT lnP (X) or G(X2)−G(X1) = −kBT ln(P (X2)P (X1)

)For example, whenP (X) is a Gaussian distribution

P (X) ∝ e−aX2

then the free energy curveG(X) is a harmonic potential

G(X) = −kBT ln e−aX2

+ C = kBTaX2 + C

whereC is just a constant coming from the normalization factor ofP (X). It is verystraightforward to calculatedP (X) from simulations. The umbrella sampling methoddiscussed in the previous section can be employed for high energy (less probable) re-gions alongX.

Chapter 7

Time-dependent QuantumTheory

Hamiltonian :H = H0 + V (q, t) H0 (time-independent) :

H0φn(q) = Enφn(q) ⇒ steady-state:φn(q)e−iEnt/h

Time-dependent Schrodinger equation:

ih∂

∂tψ(q, t) = Hψ(q, t) (7.1)

Expandψ(q, t) by φn (= complete orthonormal set)

ψ(q, t) =∑n

cn(t)φn(q)e−iEnt/h

Enter into (7.1), multiplyφk and integrate over the coordinates (i.e., operate〈φk|× ),and use orthogonality〈φk|φn〉 = δkn, to get

d

dtck(t) = − i

h

∑n

eiωkntVkn(t)cn(t) (7.2)

whereVkn(t) =∫φk(q)∗V (q, t)φn(q)dq = 〈k|V (t)|n〉 andωkn = (Ek−En)/h, i.e.,

hωkn is the transition energy forn↔ k.

Illustration: 2-level system It is often useful to look at simplest examples, in thiscase transition between two states where (7.2) resuces to

d

dt

[c1(t)c2(t)

]= − i

h

[V11(t) e−iω12tV12(t)e+iω12tV21(t) V22(t)

] [c1(t)c2(t)

]We introduce the following simplifying assumptions

41

42 CHAPTER 7. TIME-DEPENDENT QUANTUM THEORY

• V11 = V22 = 0 (V = interaction among different states)

• V12(t) = V21(t) = v (const.) • ω12 = 0 (orE1 = E2)

Thend

dtc1 = − i

hvc2,

d

dtc2 = − i

hvc1

This set of differential equations can be easily solved by taking the sum and differenceof the two. With the initical conditionc1(0) = 1 andc2(0) = 0,

c1 + c2 = e−ivt/h, c1 − c2 = e+ivt/h

⇒ c1 = cos(vt/h), c2 = i sin(vt/h)

Therefore,c1 andc2 show alternating oscillation (or ‘resonance’) along time with theoscillation period∝ v−1. Note that the larger interaction (v) causes the faster oscilla-tion.

7.1 Perturbative expansion

Now we come back to the general form of (7.2), which can be rewritten in a matrixform as

d

dtc(t) = − i

hW(t) · c(t) (7.3)

where the matrix element is given by[W(t)]kn = eiωkntVkn(t). It can be integratedformally

c(t) = c(0)− i

h

∫ t

0

W(τ) · c(τ)dτ (7.4)

Of course, the problem has not been solved because the right hand side contains theunknownc(τ). We may proceed by putting the whole of right hand side into thec(τ)in the integrand to get

c(t) = c(0)− i

h

∫ t

0

W(τ) · c(0)dτ +(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′W(τ) ·W(τ ′) · c(τ ′)

(7.5)Repeating the analogous procedure, we get a series expansion

c(t) = c(0)− i

h

∫ t

0

W(τ) · c(0)dτ +(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′W(τ) ·W(τ ′) · c(0)

+(−ih

)3 ∫ t

0

dτ

∫ t

0

dτ ′∫ t

0

dτ ′′W(τ) ·W(τ ′) ·W(τ ′′) · c(0) + · · ·

(7.6)

7.2. FIRST-ORDER PERTURBATION 43

7.2 First-order perturbation

When the interactionV is a weak perturbation, we might neglect the higher order termsin (7.6)

c(1)(t) = c(0)− i

h

∫ t

0

dτW(τ) · c(0) (7.7)

This is called the first-order perturbation treatment. Now assume that the system waspurely in them-th state at timet = 0, i.e.,cn(0) = δnm. The interactionV may inducetransition to the other states, say|k〉, whose coefficient will be according to (7.7),

c(1)k (t) = δkm −

i

h

∫ t

0

dτ∑n

Wkn(τ)cn(0) = − ih

∫ t

0

dτWkm(τ) (7.8)

Therefore, the transition probability from state|m〉 to |k〉 is

P(1)k (t) = |c(1)

k (t)|2 =1h2

∣∣∣∣∫ t

0

dτVkm(τ)eiωkmτ∣∣∣∣2 (7.9)

We may proceed further by assuming that the interactionVkm is a constant indepen-dent of time. In other words, we consider a situation where the constant perturbationis switched on at timet = 0. The system, originally populated among the unper-turbed states underH0, will now start to mix via the transitions along time. For thetime-independentVkm, (7.9) will be

P(1)k (t) = |Vkm|2

sin2(ωkmt/2)(hωkm/2)2

(7.10)

Since the initial state|m〉 is given, we may look atP (1)k (t) as a function ofEk: It is

a sharply peaked functin atEm having the width∼ 2πh/t and height∼ (t/h)2. Thepeak atEm means that the transition is most probable when the energy is conserved.Since the width is proportional to1/t, the function is broad in the short time, whichmeans that the requirement of energy conservation is less strict in the shorter time. Thisis a manifestation of the uncertainty principle between the energy and time.

7.3 Second-order perturbation

c(2)(t) = c(1)(t) +(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′W(τ) ·W(τ ′) · c(0)

Assume :cn(0) = δnm (as before)

[W(τ) ·W(τ ′) · c(0)]k =∑n

∑l

Wkn(τ)Wnl(τ ′)cl(0) =∑n

Wkn(τ)Wnm(τ ′)

⇒ c(2)k (t) = c

(1)k (t) +

(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′∑n

Vkn(τ)Vnm(τ ′)eiωknτeiωnmτ′

Transition|m〉 → |k〉 via intermediate states|n〉


Examples:

• Raman scattering

• Bridge-mediated electron transfers

Important when 1st-order (direct) transitions are forbiddenVkm = 0

7.4 Schrodinger vs Interaction Pictures

Schrodinger picture:

ih∂

∂t|ψS(t)〉 = H(t)|ψS(t)〉

Integrate:

|ψS(t)〉 = |ψS(0)〉 − i

h

∫ t

0

dτH(τ)|ψS(τ)〉

Perturbative expansion (as in section 7.1): (we omit subscriptS)

|ψ(t)〉 = |ψ(0)〉 − ih

∫ t

0

dτH(τ)|ψ(0)〉

+(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′H(τ)H(τ ′)|ψ(0)〉

+ · · ·

But, dealing with the operatorsH(t) is rather tedious

⇒ Interaction picture is often more convenient [H = H0 + V ]

7.5 Interaction picture

Definition:|ψI(t)〉 ≡ eiH0t/h|ψS(t)〉

Eq of motion:∂

∂t|ψI(t)〉 = − i

hVI(t)|ψI(t)〉

whereVI(t) ≡ eiH0t/hV e−iH0t/h

∂

∂t|ψI〉 =

i

hH0e

iH0t/h|ψS〉+ eiH0t/h∂

∂t|ψS〉

= − iheiH0t/hV |ψS〉 = − i

heiH0t/hV e−iH0t/h|ψI〉

7.6. TIME-ORDERED EXPONENTIAL 45

Perturbative expansion:

|ψI(t)〉 = |ψI(0)〉 − ih

∫ t

0

dτVI(τ)|ψI(0)〉

+(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′VI(τ)VI(τ ′)|ψI(0)〉+ · · ·

(Compare with the Schrodinger picture:H(t)→ VI(t) )

7.6 Time-ordered Exponential

Assume:|ψI(0)〉 = |φm〉 (= |m〉).The amplitudes of other states|k〉 (6= |m〉) at timet are

ck(t) = 〈k|ψI(t)〉

= 〈k|1 +(−ih

)∫ t

0

dτVI(τ) +(−ih

)2 ∫ t

0

dτ

∫ t

0

dτ ′VI(τ)VI(τ ′) + · · · |m〉

≡⟨k

∣∣∣∣exp+

[−ih

∫ t

0

dτVI(τ)]∣∣∣∣m⟩ (time-ordered exponential)

This can also be expressed (formally) using “Dyson’s time-ordering operator” T

TA(t1)B(t2) =A(t1)B(t2) (t1 > t2)B(t2)A(t1) (t1 < t2)

exp+

[−ih

∫ t

0

dτVI(τ)]

= T∞∑n=0

[−ih

∫ t

0

dτVI(τ)]n

= T exp[−ih

∫ t

0

dτVI(τ)]

7.7 Fermi’s golden rule

Assume: • time-independentVkm, • ck(0) = δkm (or |ψ(0)〉 = |m〉 )

⇒ P(1)k (t) = |c(1)

k (t)|2 = |Vkm|2sin2(ωkmt/2)(hωkm/2)2

The total probability of transition from|m〉 is

PT ≡∑k 6=m

P(1)k (t)

=∑k

|Vkm|2sin2(ωkmt/2)(hωkm/2)2

=∫dEρ(E)|V∗m(E)|2 sin2((E − Em)t/2h)

((E − Em)/2)2


whereρ(E) =∑k

δ(E − Ek) is the density of states.

In the limit of t→∞:

PT →∫dEρ(E)|V∗m(E)|2 2πt

hδ(E − Em) =

2πthρ(Em)|V∗m|2

Transition rate: wT =2πh|V∗m|2ρ(Em) · · · Fermi’s golden-rule

7.7.1 State-to-state form

In the limit t→∞P

(1)k (t)→ |Vkm|2

2πthδ(Ek − Em)

State-to-state transition rate:wkm =2πh|Vkm|2δ(Ek − Em)

By summing over the final states,wT is recovered (as a matter of course...)∑k

wkm =∫dEkρ(Ek)wkm =

2πh|V∗m|ρ(Em) = wT

Note : Ast→∞:

sin2(·)(·)2

→ δ(Ek − Em) ∼ Energy conservation

Energy width ( = uncertainty)→ smaller. This represents the uncertainty principle (E ↔ t ).

7.7.2 Periodic interaction

Let us consider an interaction periodic in time

V (t) = Ue±iωt

(e.g. semiclassical theory of light-matter interaction)

c(1)k (t) =

i

h

∫ t

0

dτVkm(τ)eiωkmτ =i

h

∫ t

0

dτUkmei(ωkm±ω)τ

We can use the previous results straightforwardly, by replacingVkm → Ukm andωkm → ωkm ± ω• Total transition rate:

wT =2πh|Ukm|2ρ(Em ∓ hω)

• State-to-state form:

wkm =2πh|Ukm|2δ(Ek − Em ± hω)

Namely, the energy conservation condition is just ‘dressed’ by the photon energyhω.

Chapter 8

Light-Matter Interaction

8.1 Electromagnetic wave

(outline)

1. Maxwell Eq with vector (A) and scalar (φ) potentials

2. Vacuum (no charge, no current)ρ = 0, i = 0, Coulomb gauge∇A = 0

3. ⇒Wave Eq :1c2∂2A∂t2

= ∇2A⇒ Plane waveA = A0ε cos(k · r− ωt)

4. (Energy from fields amplitude) = (Photon density)×(Energy quantum)⇒ A0 = 2c(2πhN/ωV )1/2

8.1.1 “Minimal” Electromagnetic Interaction

(charged particles)

H =1

2m

(p− q

cA)2

+qφ → classical eq of motion with Lorenz force

F = q(E + v ×B/c)Using :p→ ih∇ ,∇ ·A = 0 (Coulomb gauge)

H = H0 + V, V = − q

mc(A · p) +

q2

2mc2A ·A

(“weak-field approximation” neglects the 2nd term

8.1.2 Absorption and Emission Spectra

(1st order processes)

V = − q

mc(A · p) = − q

mcA0 cos(k · r− ωt)ε · p

≡ U(k)e−iωt + U(−k)eiωt[U(k) ≡ − qA0

2mceik·rε · p

](periodic interaction)

47

48 CHAPTER 8. LIGHT-MATTER INTERACTION

⇒Fermi’s golden-rule :|m〉 → |k〉

wkm =2πh|Ukm(k)|2δ(Ek − Em − hω) (Absorption)

+2πh|Ukm(−k)|2δ(Ek − Em + hω) (Emission)

(Absorptionor Emission⇐ sign ofEk − Em)

8.2 Dipole Approximation (long wavelength)

Visible lights (electronic transitions) :λ =400∼ 700 nm dimension of molecules

(even longer wavelength for infrared and microwave lights)⇒ eik·r ' 1 (|k| = 2π/λ)

Then, Ukm ' −qA0

2mc〈k|ε · p|m〉

Using : p =im

h[H0, r] [H0, ri] = [

∑I

P2I

2MI+∑j

p2j

2m + V (r,R), ri]

= 12m [p2

i , ri] = himpi

Ukm = − qA0

2mcim

hε · 〈k|[H0, r]|m〉 = − qA0

2mcim

hε · 〈k|r|m〉(Ek − Em)

From theδ functions inwkm , Ek − Em = ±hω (+ : absorption,− : emission)

⇒ Ukm = ∓ iA0ω

2cε · 〈k|qr|m〉 ∝ transition dipole moment

So far, we have treated one particle (with chargeq ).By considering all nuclei and electronsin molecular systems,

V = −1cA · (

∑I

ZIMI

PI −e

m

∑i

pi)

We arrive at (Note : all calculations were linear)

Ukm = ∓ iA0ω

2cε · 〈k|(

∑I

ZIRI − e∑i

ri)|m〉 = ∓ iA0ω

2cε · µkm

µkm : transition dipole of the molecular system

8.2.1 Simpler derivation ofUkmWe get the sameUkm (more easily) starting fromV = −µ ·E with

E = −1c

∂A∂t

=ω

cA0ε sin(k · r− ωt) =

ωA0

2icε(ei(k·r−ωt) − e−i(k·r−ωt))

(Coulomb gauge∇φ = 0 )

8.3. DENSITY OF STATES OF PHOTON FIELD 49

8.3 Density of states of photon field

Having specified the states (|m〉 → |k〉 ) of the molecular system, we still need tosum over the photon field states

wabs/em(m→ k) =∫wkmρphoton(hω)d(hω)

Insertwkm of page 2 :

wabs(m→ k) =2πh|Ukm(k)|2ρphoton(Ek − Em)

wem(m→ k) =2πh|Ukm(−k)|2ρphoton(Em − Ek)

8.3.1 Calculation ofρphoton(hω)

1. Number of statesN in wave vectork : dN = ( L2π )3dkPeriodic boundary condition in lengthL :

eikx(x+L) = eikxx⇒ kxL = 2πnx (nx = 0,±1,±2, · · · )

2. Use polar coordinatedk = k2dkdΩ andk = ω/c : dN =V

(2π)3

ω2

c3dωdΩ

3. dρphoton(hω) =dN

d(hω)=

V

(2πc)3

ω2

hdΩ (Differential) density of photon

states havingk directed todΩ

Thus, (differential) emission rate towarddΩ :

dwem(m→ k)dΩ

=2πh|Ukm|2

dρphoton(Ek − Em)dΩ

=Nω3

2πhc3|ε · µkm|2

used|Ukm| = (A0ω/2c)|ε · µkm|, A0 = 2c(2πhN/ωV )1/2, hω = Ek −Em• |ε · µkm| = |µkm| sin θ (θ ≡ angle betweenk andµkm)

• Integration over the solid angledΩ∫dΩ sin2 θ =

∫ 2π

0dϕ∫ π

0sin θdθ sin2 θ =

8π3

⇒ Total emission rate

wem(m→ k) =4N3h

(ωc

)3

|µkm|2

• However, it is not correct to havewem → 0 for N → 0, since in reality, we have“spontaneous emissions”.

• To remedy this, we just need to replaceN byN + 1, which is indeed justifiedvia quantization of radiation(photon) field.


• Quantization of radiation field: Photon state|n〉 =1√n!

(b†)n|0〉

〈n− 1|b|n〉 =√n 〈n+ 1|b†|n〉 =

√n+ 1

(absorption) (emission)〈1|b†|0〉 = 1 6= 0⇒ spontaneous emission into the photon vacuum is possible.(On the other hand, we have no “spontaneous absorption” sinceb|0〉 = 0 )

8.3.2 Natural lifetime of excited states

of an excited state|m〉 ⇐ summing over all|k〉 in lower energyEk < Em :

1τrad(Em)

=∑

Ek<Em

wem(m→ k)|N=0 =∑

Ek<Em

43h

(|ωkm|c

)3

|µkm|2

8.3.3 Rate of photo-absorption

(Similarly to the emission case)

wabs(m→ k) =4

3hN(ωc

)3

|µkm|2

wabs → 0 asN → 0 : no “spontaneous absorption”

8.4 Miscellaneous

8.4.1 Einstein’s A and B factors

Photon field density of states (or energy)

• Many others use black-body Planck distribution (temperature dependent)or discuss without explicit form of the density of states.(Indeed, the A, B factors are defined from the photon-field independent partofthe transition rate.)

8.4.2 Oscillator strength

fkm ≡2mω3he2

|µkm|2 (dimensionless)

Defined as relative intensity ton = 0→ 1 transition of 3D harmonic oscillator (which givesf = 1 ... (*)).

Exercise : Derive (*) via the following two routes:

1. Using creation-annihilation operator

2. Carrying out Gaussian integration of the wavefunctions representation

8.4. MISCELLANEOUS 51

Answer : (3D H.O. H = 12mp2 + mω2

2 |x|2, λ ≡ mω/h )

1. b = (mω/2h)1/2(x + ip/mω), b† = (mω/2h)1/2(x − ip/mω) ⇒ x =(mω/2h)1/2(b+ b†)/2

Usingb|n〉 =√n|n− 1〉, b†|n〉 =

√n+ 1|n+ 1〉 ⇒ 〈1|x|0〉 = (h/2mω)1/2

Thus,|〈1|er|0〉|2 = 3× he2

2mω (where3× from x, y, z components)

2. φ0 = (λ/π)1/4 exp(−λx2/2), φ1 =√

2λ(λ/π)1/4x exp(−λx2/2)

⇒∫dxφ0xφ1 =

√2/πλ

∫dxx2e−λx

2= · · ·

8.4.3 Correspondence with experiments

Beer-Lambert Law :Il = I0e−αCl α ... absorption coefficient

C ... molar concentrationIntegrated absorbanceA ≡

∫α(ν)dν

Considering the decay of photon field energy, we get (skipping details)

A =hνkmc

LB =4π2νkmL

3hc|µkm|2

B ... Einstein’s B-coeff.L ... Avogadro number

νkm ... central peak frequency of the|m〉 → |k〉 absorption band

(or in the derivation, representativeν when approximating∫ α(ν)

ν dν ' 1ν

∫α(ν)dν )

8.4.4 1st correction to long-wavelength approx.

Dipole Approx :eik·r ' 1 (|k| = 2π/λ)To 1st-order :eik·r = 1 + k · r + · · ·

Ukm = − eA0

2mc〈k|eik·rε · p|m〉

' − iA0ωkm2c

ε · µkm︸︷︷︸− i2 A0(k× ε)︸︷︷︸ ·Mkm +14e

A0ωkmc

ε︸︷︷︸ ·Qkm · k(dipole approx.) H E

• M = e2mc (r× p) ... magnetic dipole

• Qij = (eri)(erj) ... electric quadrupole tensor(ri = x, y, z)

Symmetry :• µ ∼ x, y, z , •M ∼ Rx, Ry, Rz, (rotation) •Q ∼ xy, x2, etc.

⇒ different symmetry for each term


8.5 Franck-Condon Factor

Molecular wavefunction (adiabatic approx.) Φε,ν(r,R) = χε,ν(R)ϕε(r;R)Quantum number

ε ... electronicν ... nuclearTransition dipole for(ε, ν)→ (ε′, ν′) :

µε′ν′,εν =∫dR∫drχε′ν′ϕε′µχενϕε =

∫dRχε′ν′

∫drϕε′µϕε︸︷︷︸χεν

= 〈χε′ν′ |µε′ε(R)|χεν〉 ≡ µε′ε(R)

Expandµ around a (particular) nuclear configurationR0

(usually minimum of the adiabatic potential)

µε′ε(R) = µε′ε(R0) +(∂µε′ε∂R

)R=R0

(R−R0) + · · ·

Then,µε′ν′,εν = µε′ε(R0) · 〈χε′ν′ |χεν〉︸︷︷︸ + (non-Condon terms)

(Franck-Condon factor)

In many cases, large Franck-Condon overlap at “vertical transitions”⇔ Nuclei don’t move during fast electronic transitions

(semi-classical interpretation)

Chapter 9

Time Correlation Functions

wkm =2πh|Ukm|2δ(Ek − Em ± hω)

Molecular systems :|m〉 = |i, ν〉 → |k〉 = |f, ν′〉 · · · |electrons, nuclei〉

Ufν′,iν =∫dRχfν′(R)

∫drϕf (r;R)U(r,R)ϕi(r;R)︸︷︷︸χiν(R)

≡ 〈χfν′(R)| Ufi(R)︸︷︷︸ |χiν(R)〉R

Ufi(R) ≡ 〈ϕε′(r;R)|U(r,R)|ϕε(r;R)〉rSuppose : we cannot specify the final nuclear quantum stateν′

(e.g., have not sufficient energy resolution, or not interested)

w(f ← iν) =∑ν′

wfν′,iν

( Note : normally, vib. rot. energy∼ kBT , electronic energy kBT )And : thermal average over the initial nuclear statesν

w(f ← i) =∑ν

Piνw(f ← iν), Piν = e−βEiν/Zi

(Boltzmann distribution)• Partition function : (⇒ normalization

∑ν Piν = 1 )

Zi =∑ν

e−βEiν =∑ν

〈ν|e−βHi |ν〉 = Tr(nuc)[e−βHi ]

BO approx. He(r;R)ϕi(r;R) = Wi(R)ϕ(r : R)Hiχiν(R) = Eiνχiν(R) , whereHi ≡ TN +Wi(R)

• Density operator for nuclear states on adiabatic potentialWi(R)

ρi ≡ e−βHi/Zi, Zi = Tr(nuc)[e−βHi ]

53

54 CHAPTER 9. TIME CORRELATION FUNCTIONS

• Thermal average (onWi(R)) of a quantityA(R)

〈A〉 =∑ν

Piν〈ν|A|ν〉 =∑ν

e−βEiν

Zi〈ν|A|ν〉 =

∑ν

〈ν|e−βHi

ZiA|ν〉

= Tr(nuc)[ρiA]

9.0.1 Time-dependent form

(Kubo-Toyozawa)Using : δ(ω) = 1

2π

∫∞−∞ eiωtdt , (and henceδ(hω) = 1

hδ(ω) )

w(f ← iν) =2πh

∑ν′

|Ufν′,iν |2δ(Efν′ − Eiν ± hω)

=1h2

∑ν′

|〈ν′|Ufi|ν〉|2∫ ∞−∞

dte−iEfν′ t/he+iEiνt/he∓iωt

=1h2

∑ν′

∫ ∞−∞

dt〈ν|U∗fie−iHf t/h|ν′〉〈ν′|UfieiHit/h|ν〉e∓iωt

=1h2

∫ ∞−∞

dt〈ν| U∗fie−iHf t/hUfieiHit/h︸︷︷︸ |ν〉e∓iωt≡ A (for brevity)

w(f ← i) =∑ν

Piνw(f ← iν) =1h2

∫ ∞−∞

dtTr(nuc)[ρiA]e∓iωt

Application - 1 : Vibration / rotation spectraIn the same electronic state :Hf = Hi = H

⇒ Ufi = Uii ∝ µ (dipole moment)Vibration / rotation spectra (absorption)

σ(ω) ∝∫∞−∞ dtTr(nuc)[ρµ∗ e−iHt/hµe+iHt/h︸︷︷︸]eiωt

µ(−t) (Heisenberg rep.)=∫∞−∞ dt〈µ∗(0)µ(−t)〉eiωt

(9.1)

• Vib-rot spectra⇔ TCF of vib-rot (thermal) motionApplication - 2 : Electronic spectra, Nonadiabatic transitions

1. Gaussian Wavepacket method

2. Cumulant Expansion method

9.0.2 Gaussian Wavepacket

χpt,qt(q, t) = exp[i

hαt(q − qt)2 +

i

hpt(q − qt) +

i

hγt]

9.1. CUMULANT EXPANSION METHOD 55

• qt = 〈q〉, pt = 〈p〉 : average position and momentafollow classical eq of motion :xt = ∂H/∂pt, pt = −∂H/∂xt

• αt, γt : width and phase (time dependent)αt = −(2/m)α2

t − Vxx/2, γt = ihαt/m+ ptxt − E• Exact on quadratic potentials

V (x) = V0 + Vx(x− xt) + 12Vxx(x− xt)2

Using : Tr(nuc)[A]→ 1hn

∫dqn0 dp

n0 〈χp0,q0 |A|χp0,q0〉

w(f ← i) =1h2

∫ ∞−∞

C(t)e∓iωtdt

C(t) ∝∫dqn0 dp

n0 〈χp0,q0 |ρiU∗fie−iHf t/hUfieiHit/h|χp0,q0〉

= 1Z

∫dqn0 dp

n0 〈Φi(t− iβh)|Φf (t)〉

|Φi(t− iβh)〉 = Ufie−iHi(t−iβh)/h|χp0,q0〉

|Φf (t)〉 = e−iHf t/hUfi|χp0,q0〉

1. Propagate wavepackets|χp0,q0〉 onHi andHf

2. Calculate the overlap〈Φi|Φf 〉

3. Fourier transform

9.1 Cumulant expansion method

9.1.1 Condon Approximation :

NeglectR dependence ofUfi(R)

or take 0th term of :Ufi(R) = Ufi(R0) +

(∂Ufi∂R

)R0

(R−R0) + · · ·

⇒

w(f ← i) = 1h2 |Ufi(R0)|2

∫∞−∞ dt〈e−iHf t/he+iHit/h︸︷︷︸〉ie∓iωt

(Time ordered exponential) exp(−)[ih

∫ t0dτ∆Vi(τ)]

∆V ≡ Hf −Hi, ∆Vi(t) ≡ e−iHit/h∆V e+iHit/h

• Time ordered exponential( f(t) ≡ ) e−iHf t/he+iHit/h = exp(−)[

ih

∫ t0dτ∆Vi(τ)]

Note : Since[Hi,Hf ] 6= 0 , e−iHf t/heiHit/h 6= e−i(Hf−Hi)t/h

∂f

∂t=i

he−iHf t/h(Hf −Hi)e+iHit/h

=i

he−iHf t/he+iHit/he−iHit/h(Hf −Hi)e+iHit/h

=i

hf(t)∆Vi(t)


Integrate :f(t) = f(0) +i

h

∫ t

0

dτf(τ)∆Vi(τ)

Sequential expansion (f(0) = 1) :

f(t) = 1 +i

h

∫ t

0

dτ∆Vi(τ) + (i

h)2

∫ t

0

dτ

∫ τ

0

dτ ′∆Vi(τ ′)∆Vi(τ) + · · ·

≡ exp(−)[i

h

∫ t

0

dτ∆Vi(τ)] (⇐ Definition)

9.1.2 Note

:The original formula was

w(f ← i) =∑ν

∑ν′

Piνw(fν′ ← iν)

=2πh

∑ν

∑ν′

Piν |〈fν′|Ufi|iν〉|2δ(Efν′ − Eiν ± hω)

Condon Approx.

w(f ← i) ' 2πh|Ufi(R0)|2

∑ν

∑ν′

Piν |〈fν′|iν〉|2δ(Efν′ − Eiν ± hω)

We will obtainw(f ← i) in p.7 from the Fourier transform of this.• If we correctly evaluatew(f ← i) in p.7, it takes into account of the (thermal

average of) the Franck-Condon overlap|〈fν′|iν〉|2⇒ Quantum effects of the nuclear motion (e.g., tunneling) are accounted

9.1.3 Cumulant expansion

“average of exponential”→ “exponential of averages”

〈eiλx〉 = eiλ〈x〉c+12 (iλ)2〈x2〉c+···

Expand both sides and compare (in order ofλ ) to findCumulant average : 〈x〉c = 〈x〉

〈x2〉c = 〈x2〉 − 〈x〉2 (= variance)〈x3〉c = 〈x3〉 − 3〈x2〉〈x〉+ 2〈x〉3

Advantages:

• Average of oscillatory function〈eiλx〉→ average first, then place on exponenteiλ〈x〉c + · · ·

• Partial sum to infinite order :even if the exponent on the right-hand-side is truncated at finite order, terms upto the infinite order inλ is partially included

• In a particular case where the variablex follows Gaussian distribution (Gaussianprocess), the 3rd and higher order cumulants exactly vanish

9.2. TIME CORRELATION FUNCTIONS 57

9.1.4 Cumulant expansion of time-ordered exponential

〈exp(−)[i

h

∫ t

0

dτ∆V (τ)]〉

= exp

[i

h

∫ t

0

dτ〈∆V (τ)〉c +(i

h

)2 ∫ t

0

dτ

∫ τ

0

dτ ′〈∆V (τ ′)∆V (τ)〉c + · · ·

]Expand both sides and compare order by order (up to 2nd order) :⇒ 〈∆V (τ)〉c = 〈∆V (τ)〉 = 〈∆V (0)〉 = 〈∆V 〉 (independent of time in

steady-state)〈∆V (τ ′)∆V (τ)〉c = 〈∆V (τ ′)∆V (τ)〉 − 〈∆V 〉2 = 〈δ∆V (τ ′)δ∆V (τ)〉

( δ∆V (t) ≡ ∆V (t)− 〈∆V 〉 )

• Line-broadening function :g(t) ≡ 1h2

∫ t

0

dτ

∫ τ

0

dτ ′〈δ∆V (τ ′)δ∆V (τ)〉

• In thermal equilibrium :〈A(τ ′)A(τ)〉 = 〈A(0)A(τ ′ − τ)〉(depends only on the time intervalτ − τ ′)

[Verify using 〈...〉 = Trρ... and Heisenberg rep. ofA(t)]Changing integration variable(τ, τ ′)→ (τ ′, s ≡ τ − τ ′) : Jacobian = 1g(t) = 1

h2

∫ t0ds∫ t−s

0dτ ′〈δ∆V (0)δ∆V (s)〉 = 1

h2

∫ t0ds(t− s)〈δ∆V (0)δ∆V (s)〉

Thus,

w(f ← i) =1h2 |Ufi|

2

∫ ∞−∞

dte−gi(t)ei(〈∆V 〉i/h∓ω)t

gi(t) =1h2

∫ t

0

ds(t− s)〈δ∆V (0)δ∆V (s)〉i(correct up to 2nd-order cumulant)

• 〈∆V 〉i : average over nuclear motion on potentialVi (electronic statei)

• 〈δ∆V (0)δ∆V (t)〉i : TCF = thermal fluctuation of∆V

9.2 Time correlation functions

C(t) = 〈δA(0)δA(t)〉 : fluctuationδA(t) = A(t)− 〈A〉Classical : phase-space distribution functionf(r,p)

C(t) =∫dr∫dpf(r,p)δA(r,p; 0)δA(r,p; t)

Statistical (ensemble) average⇐ distribution of the initial conditionf(r,p)Quantum :

C(t) = Tr[ρδAe−iHt/hδAeiHt/h]

Thus,C(−t) = C∗(t)⇒ ReC(−t) = ReC(t) ... even,Im C(−t) =−Im C(t) ... odd

In classical mech.C(t) is real and even (for real quantities)C(0) = 〈δA(0)2〉 ≥ 0 (variance, fluctuation) When quantityA shows stochastic

random motion (eg, in solution phase),C(t)→ 0 ast→∞.


• δA(0) andδA(t) at larget (long time interval) lose mutual correlation, such thatbothδA(0)δA(t) > 0 and< 0 realize randomly.

• In other words, at long time interval,δA(0) and δA(t) become “statisticallyindependent”, as described byC(t)→ 〈δA(0)〉〈δA(t)〉 = 0× 0 = 0.

9.2.1 Ergodic hypothesis

Statistical (ensemble) average = average over the time

C(t) = 〈δA(0)δA(t)〉 = limT→∞

1T

∫ T

0

dt0δA(t0)δA(t0 + t)

9.3 TCF and spectral line shape

Vib. / rot. spectra :σ(ω) ∝∫∞−∞〈µ(t)µ(0)〉eiωtdt =

∫∞−∞ C(t)eiωtdt

9.3.1 Exponential TCF

C(t) = C(0)e−|γ|t⇒ σ(ω) ∝ C(0)2γ

γ2 + ω2(Lorentzian line shape)

9.3.2 Gaussian TCF

C(t) = C(0)e−λ2t2 ⇒ σ(ω) ∝ C(0)e−ω

2/4λ2(Gaussian line shape)

9.3.3 Damped-oscillating TCF

(1)C(t) = C(0)e−γ|t| · eiω0t⇒ σ(ω) ∝ 2γγ2 + (ω − ω0)2

(2)C(t) = C(0)e−λ2t2 · eiω0t⇒ σ(ω) ∝ e−(ω−ω0)2/4λ2

(Note : oscillatory factoreiω0t just introduces peak shift)[Verify by yourself (just elementary integrations!)]

9.3.4 Example of Gaussian TCF

Consider :dilute solution of dipolar moleculesShort-time motion' nearly free rotation (with angular velocityΩ)

(“inertial motion”)Dipole correlation :µ(0)µ(t) = |µ|2 cos ΩtKinetic energy of rotation =IΩ2/2 (I = inertial moment)⇒ Thermal population ofΩ : P (Ω)dΩ ∝ e−E/kBT dΩ = e−IΩ

2/2kBT dΩ(Normalization :

∫∞0P (Ω)dΩ = 1⇒ prefactor2(I/2πkBT )1/2 )

Hence,

〈µ(0)µ(t)〉 =∫ ∞

0

P (Ω)|µ|2 cos ΩtdΩ = |µ|2e−kBTt2/2I

9.4. MOTIONAL NARROWING 59

(Gaussian TCF)

9.3.5 Example of Exponential TCF

Langevin equation(Brownian motion model)

mv = −mγv +R(t)

(R(t) : random force,γ : friction coefficient)v(0)× and statistical average

〈v(0)v(t)〉 = −γ〈v(0)v(t)〉+1m〈v(0)R(t)〉

DefineC(t) ≡ 〈v(0)v(t)〉 , and assume〈v(0)R(t)〉 = 0(no correlation betweenv(0) and the random force)

d

dtC(t) = −γC(t)⇒ C(t) = C(0)e−γt

9.3.6 Example : Brownian oscillator model

Harmonic oscillator + friction + random force

mx = −mω20x−mγx+R(t)

〈x(0)x(t)〉 = −ω20〈x(0)x(t)〉 − γ〈x(0)x(t)〉+

1m〈x(0)R(t)〉

C(t) + γC(t) + ω20C(t) = 0

⇒ C(t) = C(0)(cosω1t+γ

2ω1sinω1t)e−γt/2

(ω21 ≡ ω2

0 − γ2/4 )(i) ω2

0 > γ2/4 : C(t) = damped oscillation(ii) ω2

0 = γ2/4 : C(t) = C(0)(1 + γt/2)e−γt/2

(iii) ω20 < γ2/4 : ω1 = imaginary⇒ C(t) = double exponential

[ (i) = “under-damped”, (ii, iii) = “over-damped” ]

9.4 Motional narrowing

w(f ← i) =1h2 |Ufi(R0)|2

∫ ∞−∞

e−gi(t)ei(〈∆V 〉i/h∓ω)tdt

gi(t) =1h2

∫ t

0

dτ(t− τ)〈δ∆V (0)δ∆V (τ)〉i

Assume : Exponential TCF

〈δ∆V (0)δ∆V (τ)〉 = D2e−|t|/τc


• D2 = 〈δ∆V (0)2〉 ... amplitude of fluctuation

• τc ... correlation time

⇒ g(t) =1h2 (Dτc)2(e−|t|/τc +

|t|τc− 1)

1. Large τc case (long correlation time / slow modulation)

g(t) ' D2t2/2h2 (short-time expansion int )

⇒ w(f ← i) =1h2 |U |

2

∫ ∞−∞

e−D2t2/2h2

ei(〈∆V 〉i/h∓ω)tdt =

2√π|U |2

hDexp[− h

2(ω ∓ 〈∆V 〉i)2

2D2] (Gaussian line shape)

2. Short τc case (short correlation time / fast modulation)

e−|t|/τc + |t|/τc − 1 ' |t|/τc (long time approximation int )

⇒ w(f ← i) =1h2 |U |

2

∫ ∞−∞

e−D2τc|t|/h2

ei(〈∆V 〉i/h∓ω)tdt

=|U |2

h2

2γγ2 + (ω ∓ 〈∆V 〉i)2

(Lorentzian line shape)

Width∝ γ ∝ τc (γ ≡ D2τc/h2 )

9.4.1 “motional narrowing”

:shorterτc (faster fluctuation modulation)⇒ narrower spectra

in general :near spectralpeak ∼ larget ⇒ Lorentzian shapenear spectraltail ∼ smallt ⇒ Gaussian shape

Overall shape⇐ parameterDτc/h <> 0

9.4.2 Physical interpretation

• Largeτc = slow modulation (of the nuclear configuration)

⇒ ∆V is fixed during the photo absorption/emission

⇒ distribution of∆V is directly reflected in the spectral shape

• Smallτc = fast modulation

⇒ fluctuation of∆V is averaged out in the observation time scale

Chapter 10

Dynamics in Condensed Phases

10.1 Phenomenology of Brownian Motions

mv = −m∫ t

0

Γ(t− τ)v(τ)dτ +R(t)

Γ(t) : friction kernel∼ friction depends on the past( = memory effect : delayed response of the surrounding media)

R(t) : random forceLater, GLE will be derived from a model Hamiltonian

(and thus, GLE may be time reversible)

10.1.1 Coarse graining

If : Γ(t) ' 2γδ(t) (no delay)⇒mv = −γmv(t) +R(t) (Langevin eq)

Similarly, if we look at the dynamics in (macroscopic) time scale∆tmuch larger than the (microscopic) decay time ofΓ(t) ,(i.e., “coarse-graining” in time)mv(t) = −Γmv(t) +R(t) ( Γ ≡

∫∆t

0Γ(τ)dτ '

∫∞0

Γ(τ)dτ )In this way, the time reversibilityof the (classical mechanical) dynamics is lost by

the coarse-graining of time scale.(But, GLE may be time reversible)

10.1.2 Laplace transform

[

math preparation

]

61

62 CHAPTER 10. DYNAMICS IN CONDENSED PHASES

Definition : Lf(t) =∫∞

0e−stf(t)dt = f(s) where (s > 0 )

Derivatives : Lf(t) = sf(s)− f(0) , Lf(t) = s2f(s)− sf(0)− f(0)Convolution : Lf(t) · Lg(t) = L

∫ t0f(t− τ)g(τ)dτ

proof : right hand side =∫∞

0dte−st

∫ t0f(t− τ)g(τ)dτ

variable transformation(t, τ)→ (τ, τ ′ ≡ t− τ) (Jacobian = 1)=∫∞

0dτ∫∞

0dτ ′e−s(τ+τ ′)f(τ ′)g(τ)

= ∫∞

0e−sτ

′f(τ ′)dτ ′

∫∞0e−sτg(τ)dτ = LfLg

Useful stuffs :

• Leiωt =∫ ∞

0

e−s(s−iω)tdt =1

s− iω

• Lcosωt = L(eiωt + e−iωt)/2 =12

[1

s− iω+

1s+ iω

] =s

s2 + ω2

• Lcosωt =ω

s2 + ω2

10.2 Microscopic model for GLE

10.2.1 System + Harmonic bath

H =p2s

2+ V (s) +

∑i

(p2i

2+ω2i

2x2i ) +

∑i

cixis

System-bath coupling :ci =(∂V (s, x)∂s∂xi

)pot min

Classical eqs of motion :s = ∂H

∂ps, ps = −∂H∂s , xi = ∂H

∂pi, pi = − ∂H

∂xi

⇒ s = −∂V (s)∂s

−∑i

cixi , xi = −ω2i xi − cis

1. (formally) solve the 2nd EOM forxi2. enter back to the 1st EOM fors

Laplace transform :λ2xi(λ)− λxi(0)− xi(0) = −ω2i xi(λ)− cis(λ)

xi(λ) =λ

λ2 + ω2i

xi(0) +1

λ2 + ω2i

xi(0)− ci1

λ2 + ω2i

s(λ)

Back transformationxi(t) = xi(0) cosωit+ xi(0)

ωisinωit− ci

ωi

∫ t0

sinωi(t− τ)s(τ)dτPartial integration of the last integral∫ t

0sinωi(t− τ)s(τ)dτ = [ 1

ωicosωi(t− τ)s(τ)]t0 − 1

ωi

∫ t0

cosωi(t− τ)s(τ)dτEnter into EOM forss = −∂V (s)

∂s −∑i(ciωi

)2∫ t

0cosωi(t− τ)s(τ)dτ − s(t) + s(0) cosωit

+R(t)

R(t) ≡ −∑i cixi(0) cosωit−

∑iciωixi(0) sinωit

Define friction kernel :Γ(t) ≡∑i(ciωi

)2 cosωit

10.3. FLUCTUATION-DISSIPATION THEOREM 63

⇒ GLE form :

s = −∂V (s)∂s

+ Γ(0)s(t)−∫ t

0

Γ(t− τ)s(τ)dτ − s(0)Γ(t) +R(t)

For harmonicV (s) = Ω2

2 s2 : −∂V (s)

∂s +Γ(0)s(t)⇒−(Ω2 − Γ(0)︸︷︷︸)s(t) ≡

−Ω2eff

ie, frequency shift (potential softening) due to friction

10.3 Fluctuation-dissipation theorem

〈R(0)R(t)〉 = kBT Γ(t)TCF of random force = friction kernel× temperature

(Both stems from the medium motion)For the harmonic bath system,

• 〈xi(0)xj(0)〉 = 0 for (i 6= j) : bath modes are independent

• 〈xi(0)xi(0)〉 = 0 position and velocity are (locally) independent

• 〈ω2i

2 xi(0)2〉 = kBT2 : equipartition theorem

Thus,〈R(0)R(t)〉 =

∑i

c2i 〈xi(0)〉2 cosωit

= kBT∑i

(ciωi

)2 cosωit = kBT Γ(t)

10.4 Matrix partitioning method

Multidimensional potential :V (x)Expand around the minimumx0 : (ie,

(∂V∂x

)x=x0

= 0 )

V (x) = V (x0) + 12Ω2(x− x0)2 + · · · [ Ω2 ≡

(∂2V∂x2

)x0

]

Off-diagonal(Ω2)ij = coupling betweenxi andxj(Note : diagonalization ofΩ2 ⇒ normal mode analysis)

Suppose :

• we are only interested in small number ofxi (i = 1, 2, · · · , n ) out of totalNdegrees of freedom. (n < N )

• Now we denote the rest ofxi (i = n+ 1 , · · ·,N ) by yi


10.4.1 Matrix partitioning

d2

dt2

[xy

]= −

[Ω2xx Ω2

xy

Ω2yx Ω2

yy

] [xy

]⇒ x = −Ω2

xxx−Ω2xyy

y = −Ω2yxx−Ω2

yyy

Similarly as before, (1) formally solve fory, (2) enter back into eq forxs2y(s)− sy(0) + y(0) = −Ω2

yxx(s)−Ω2yyy(s)

y(s) = (s21 + Ω2yy)−1(sy(0)− y(0)−Ω2

yxx(s)y(t) = cos Ωyyt·y(0)−Ω−1

yy sin Ωyyt·y(0)−Ω−1yy

∫ t0

sin Ωyy(t−τ)·Ω2yxx(τ)dτ

Partial integration, and define random force and friction kernelR(t) ≡ Ω2

xy cos Ωyyt · y(0)−Ω2xyΩ

−1yy sin Ωyyt · y(0)

Γ(t) ≡ Ω2xyΩ

−2yy cos Ωyyt ·Ω2

yx

⇒ GLE form :

x = −Ω2effx(t)−

∫ t

0

Γ(t− τ)x(τ)dτ − Γ(t)x(0) + R(t) (Verify

Fluctuation-dissipation theorem :〈R(0)R(t)〉 = kBT Γ(t) )• Set up models forR(t)⇒ Stochastic trajectory methods

10.5 Projection operator methods (1)

The division intox andy in the previous section is also obtained by applying projection operator matrices

P ≡[

1n 00 0

]and Q ≡ 1N −P =

[0 00 1N−n

](Note :P2 = P, Q2 = Q ∼ projection operator)

Starting from original (fullN dim) : x = −Ω2x = −Ω2(P+Q)x(P×)⇒ Px = −(PΩ2P)(Px)− (PΩ2Q)(Qx)(Q×)⇒ Qx = −(QΩ2P)(Px)− (QΩ2Q)(Qx)

(DefinePx ≡ xP etc.)⇒

xP = −Ω2PPxP −Ω2

PQxQxQ = −Ω2

QPxP −Ω2QQxQ

We may try to define more general projection matrices to extract physical variablesof specific interests.

10.6 Projection operator methods (2)

Projector onto a (finite) target spaceφi (i = 1, 2, · · · , n)

P ≡n∑i=1

|φi〉〈φi|, Q ≡ 1− P =∞∑

i=n+1

|φi〉〈φi|

Time-dependent Schrodinger eq :ψ = − ihHψ = − i

hH(P + Q)ψ

P ψ = − ih(P HP )Pψ + (P HQ)Qψ

Qψ = − ih(QHP )Pψ + (QHQ)Qψ

⇒ ψP = − ih (HPPψP +HPQψQ)

ψQ = − ih (HQPψP +HQQψQ)

10.7. PHASE SPACE DISTRIBUTION FUNCTION 65

Formal solution of the 2nd line (Laplace Tr.)sψQ(s)− ψQ(0) = − i

hHQP ψP (s)− ihHQQψQ(s)

ψQ(s) = 1s+iHQQ/h

ψQ(0)− ihHQP ψP (s)

⇒ ψQ(t) = e−iHQQt/hψQ(0)− ih

∫ t0e−iHQQ(t−τ)/hHQPψP (τ)dτ

Usually, we assume that the initial wavefunctionψ(0) is in the target space,in other words,ψQ(0) = Qψ(0) = 0Then the eq forψP becomes

∂

∂tψP (t) = − i

hHPPψP (t) + (

i

h)2

∫ t

0

HPQe−iHQQ(t−τ)/hHQPψP (τ)dτ

• 1st term = evolution due toHPP

• 2nd term = transition from and toQ-space

(Note : Green’s function representation⇒ damping theory )We can also carry out similar projection for the Liouville eq (cf Chap 7)∂

∂tρ = −iLρ (Liouville operatorLA ≡ 1

h [H,A] )

This leads to theMaster equation formalism of the density matrixρ

10.7 Phase Space Distribution Function

Chapter 11

Theory of Liquids

67

68 CHAPTER 11. THEORY OF LIQUIDS

Chapter 12

Reaction Rate Theory

12.1 Definition in phase space (Classical mechanics)

Phase space :(p,q) = (pi, qi), i = 1, 2, · · · , N (N degrees of freedom)Dividing surface :f(q) = 0 (N -1 dimension)

Flux through the div surface :F (p,q) = δ[f(q)]∂f(q)∂q

pm

Characteristic function :

χ(p,q) =

1 · · · trajectory passing(p,q) is reactive0 · · · else

(“reactive” = end up in the product state)Note : to determineχ, complete info of classical trajectories is required

⇒ we need some approximations (e.g., TST)

12.2 Microcanonical ratek(E)

Constant energyE (⇒ δ[E −H(p,q)] )

k(E) =h−N

∫dp∫dqδ[E −H(p,q)]F (p,q)χ(p,q)

h−N∫dp∫Rdqδ[E −H(p,q)]

•∫Rdq ≡ integration over the reactant configuration

• denominator = density of states in reactant≡ ρR(E)

• numerator×h ≡ N(E) : cumulative reaction probability (dimensionless)

⇒ k(E) =N(E)hρR(E)

(Relation with the RRKM theory will be discussed later)

69

70 CHAPTER 12. REACTION RATE THEORY

12.3 Canonical ratek(T )

Constant temperatureT ( β ≡ 1/kBT )

k(T ) = 1QR

∫dEk(E)ρR(E)e−βE ( QR ≡

∫dEρR(E)e−βE )

(Thermal average ofk(E)) (partition function inR )

⇒ k(T ) = Q−1R h−N

∫dp∫dqe−βH(p,q)F (p,q)χ(p,q)

12.3.1 Note

: bothk(E) andk(T ) do not depend on the choice off(p,q)(as far asf(p,q) is set to make sense...)

• From theLiouville’s theorem (= continuity of the phase space distribution func-tion), the net flux in a closed surface vanishes.

• Thus, flux across anytwo dividing surfacesf1(q) = 0 andf2(q) = 0 are thesame by closing them at sufficiently far away from the relevant configurationspace region

12.4 Transition state theory (TST)

Approximate the characteristic functionχ(p,q)(which is supposed to contain complete infoof the classical trajectories)

• Fundamental assumption of TST

By properly defining the dividing surfacef(q) = 0, trajectories passing throughit (toward the product region) areall “reactive”

(ie, neglect any “recrossings” that are against this assumption)We consider a model in which :

• (p,q) → (ps, s,pu,u) reaction coordinates and otherN -1 Dim coordi-nates

• Kinetic terms in Hamiltonian are separable

H(ps, s,pu,u) =p2s

2ms+ T (pu) + V (s,u) [ T (pu) =

N−1∑i=1

p2ui

2mui

]

• f(s,u) = s ( ie, dividing surface iss = 0 ) ⇒ F (p,q) = δ(s)psms

⇒ TST assumption is :χ(ps, s,pu,u) = θ(ps) (step function)

12.4. TRANSITION STATE THEORY (TST) 71

12.4.1 MicrocanonicalkTST(E)

Cumulative reaction probability under TST

N‡(E) =1

hN−1

∫dps

∫dpu

∫ds

∫duδ[E −H(ps, s,pu,u)]δ(s)

psms

θ(ps)

=1

hN−1

∫ ∞0

dps

∫dpu

∫duδ[E − p2

s

2ms− T (pu)− V (0,u)]

psms

Decompose theδ-function (s part andu part)δ[· · · ] =

∫dεδ[E − p2

s/2ms − ε− V (0,u0)]δ[ε− T (pu)− V (0,u) + V (0,u0)]( u0 : potential minimum ats = 0 )

N‡(E) =∫dε

∫ ∞0

dpspsms

δ[E − p2s/2ms − ε− V (0,u0)]

×

1hN−1

∫dpu

∫duδ[ε− T (pu)− V (0,u) + V (0,u0)]

• 2nd = density of statesρu(ε) for the internal energy of(u,pu)In 1st, transform the variableps → εs ≡ p2

s/2ms

N‡(E) =∫∞−∞ dε

∫∞0dεsδ[E − εs − ε− V (0,u0)]ρu(ε)

=∫∞

0dεsρu(E − εs − V (0,u0))

transform the variable toε ≡ E − εs − V (0,u0)noting thatρu(ε) is defined inε > 0

⇒ N‡(E) =∫ E−V (0,u0)

0

dερu(ε) Number of states betweenV (0,u0) (= bar-

rier top) andE for (u,pu) degrees of freedom

kTST(E) =N‡(E)hρR(E)

(RRKM theory)

12.4.2 CanonicalkTST(T )

kTST(T ) =1QR

1hN

∫dps

∫ds

∫dpu

∫du exp−βH(ps,s,pu,u) δ(s)

psms

θ(ps)

=1QR

1h

∫ ∞0

dpspsms

e−βp2s/2ms

×

1hN−1

∫dpu

∫due−β(T (pu)+V (0,u))

• 1st... = kBT/h (Verify : just a Gaussian integral)• 2nd... ≡ Q‡ue−βV (0,u0)

Q‡u ≡ 1hN−1

∫dpu

∫due−β(T (pu)+V (0,u)−V (0,u0)) Partition function for(u,pu)

at the transition state(s = 0,u = u0)

⇒ kTST(T ) =kBT

h

Q‡uQR

e−βV (0,u0)

This is the same as the “conventional” TST derived by assuming


1. Existence of the “activation complex” (X‡) , and

2. Thermal equilibrium betweenX‡ and the reactantR

However, as derived here, these assumptions are not essential for TST⇐ The samekTST(T ) is derived “dynamically” just from

• Separability ofp2s/2ms in HamiltonianH

• χ = θ(ps) (TST assumption)

• F = δ(s)psms

(Dividing surface :s = 0)

12.5 Quantum mechanical rate constant

Preface : Detailed discussion of this subject is quite involved, so we only summarizethe framework/outline of the representative two theories.

1. Flux-operator formalism (W H Miller et al.)

k(T ) = Q−1R Tr[e−βH F (p, q)P ]

• Flux operator :F (p, q) ≡ δ[f(q)]∂f(q)∂q

pm

• P : projection operator corresponding toχ(p,q) in the classical limit

(For example, projection to reactant states having positive momentum towardreactive collision in gas-phase reactions)

Further assumptions corresponding to the classical TST, e.g.,P → θ(ps), woulddefine “Quantum TST” (but seems not well-established...)

2. Time correlation function formalism (T Yamamoto)Based on “linear response theory for internal thermal forces” (Kubo et al.)

k(T ) =Q

βQR

∫ ∆t

0

dt

∫ β

0

dλ〈NRNR(t+ ihλ)〉

• NR : projection operator defining the reactant (or product) state

e.g., For|ψR〉 → |ψP 〉 ,NR = |ψR〉〈ψR| (with 〈ψR|ψP 〉 = 0 )

• NR = (i/h)[H,NR]

• ∆t (plateau time)

microscopically long (> decay time of∫ β

0dλ〈NRNR(t+ ihλ)〉 )

macroscopically short such that〈∆NR(t+ ∆t)〉 − 〈∆NR(t)〉

∆t' d〈NR〉

dt

12.6. CORRECTION FROM TST 73

12.6 Correction from TST

Especially in condensed phase, “recrossings” may become significantWriting the exact reaction rate ask = κkTST ,κ(= k/kTST) is called “transmission coefficient”

1. Grote-Hynes theory

2. Kramers limit

12.6.1 Grote-Hynes theory

Describe the microscopic dynamics near the barrier top (TS) by GLE

s(t) = ω2b,eqs(t)−

∫ t

0

dτζ(t− τ)s(τ) +R(t)

• ω2b,na = −(∂2H/∂s2)s=0 : “non-adiabatic” barrier frequency

• ω2b,eq = ω2

b,na + ζ(0) : equilibrium/adiabatic barrier frequency

cf. harmonic model

H = p2s/2− ω2

b,nas2/2 +

∑i

(p2i /2 + ω2

i x2i /2) + s

∑i

cixi

= p2s/2− ω2

b,nas2/2 +

∑i

p2i /2 +

∑i

ω2i (xi + cis/ω

2i︸︷︷︸)2/2− c2i s2/2ω2

i

If the bathsxi always satisfyxi+cis/ω2i = 0 , ie, adiabatically followtheir minima along each value ofs

, then the “effective” potential forswould look like− 12 (ω2

b,na+c2i /ω2i )s2 = − 1

2 (ω2b,na+

ζ(0))s2 ≡ − 12ω

2b,eqs

2

Laplace tr.

λ2s(λ)− λs(0)− s(0) = ω2b,eq s(λ)− ζ(λ)(λs(λ)− s(0)) + R(λ)

s(λ) =(λ+ ζ(λ))s(0) + s(0) + R(λ)

λ2 − ω2b,eq + λζ(λ)

inverse transformation :s(t) =∑

resλ

eλts(λ)

Grote-Hynes equation :λ2r − ω2

b,eq + λr ζ(λr) = 0From its solutionλr , the transmission coefficient is given by

κGH =λrωb,eq

(=ωb,eq

λr + ζ(λr))

For details, see, eg, Gertner, Wilson, and Hynes, J Chem Phys90, 3537 (1989),Appendix


12.6.2 Kramers limit

(GLE→ LE)Langevin eq limit :

s(t) = ω2b,eqs(t)− ζs(τ) +R(t) ζ ≡

∫∞0ζ(τ)dτ = ζ(λ = 0)

⇐ fast decay limit ofζ(t) , or coarse-grain the time scaleThen, GH equation :λ2 + ζλ− ω2

b,eq = 0

λ = (−ζ ±√ζ2 + 4ω2

b,eq)/2 (take+ sinceλ > 0 )

Further, in the strong friction caseζ ωb,eq ⇒ λr ' ω2b,eq/ζ

⇒ κ in the Kramers limit :κKR = ωb,eq/ζ

If we write kTST = ωR2π e−β∆G‡

⇒ kKR = κKRkTST =

1ζ

ωRωb,eq2π

e−β∆G‡ ∝ ζ−1

(originally, this was derived from Fokker-Planck equation)

12.7 κ(t) from Simulation

See the next subsection for the sign convention, ie, the reactant side is inx < 0, whichis different from the review by Hanggi et al. The TST rate is given by

kTST =〈δ[x(0)]x(0)θ[x(0)]〉

〈θ(−x)〉(12.1)

Time-dependent rate is

k+(t) =〈δ[x(0)]x(0)θ[−x(−t)]〉

〈θ(−x)〉(12.2)

Transmission coefficient is given by the plateau value of

κ(t) =〈δ[x(0)]x(0)θ[−x(−t)]〉〈δ[x(0)]x(0)θ[x(0)]〉

=〈j(0)θ[−x(−t)]〉〈j+(0)〉

=〈δ[x(0)]x(0)θ(+x) + θ(−x)θ[−x(−t)]〉

〈δ[x(0)]x(0)θ[x(0)]〉

=〈j+(0)θ[−x(−t)]〉

〈j+(0)〉+〈j−(0)θ[−x(−t)]〉

〈j+(0)〉

=〈j+(0)θ[−x(−t)]〉

〈j+(0)〉− 〈j−(0)θ[−x(−t)]〉

〈j−(0)〉= 〈θ[−x(−t)]〉+ − 〈θ[−x(−t)]〉−

(12.3)where〈j+(0)〉 + 〈j−(0)〉 = 〈δ(x)x〉 = 0 has been used. In the last line, the flux-weighted average is defined by

〈· · · 〉± =〈j±(0) · · · 〉〈j±(0)〉

=〈δ(x)xθ(±x) · · · 〉〈δ(x)xθ(±x)〉

(12.4)

12.7. κ(T ) FROM SIMULATION 75

As shown in the next subsection, using the time-reversibilty, we get

κ(t) = 〈θ[x(t)]〉+ − 〈θ[x(t)]〉− (12.5)

These averages are generated by the flux-weighted distribution function

P±(x, x) ∝ δ(x)|x|θ(±x)e−H(x,x)/kBT (12.6)

This distribution may be generated via the Metropolis Monte Carlo method by includ-ing the flux-weight factor|x| into the potential such as

P±(x, x) ∝ δ(x)θ(±x) exp(−H(x, x)− kBT ln |x|

kBT

)(12.7)

Because the position is fixed atx = 0, we just need the velocity distribution from

P±(x) ∝ θ(±x) exp[−β(m

2x2 − β−1 ln |x|

)](12.8)

12.7.1 Derivation

cf. Hanngi, Talkner and Borkovec, Rev. Mod. Phys. (1980), but use alternativeconvention for the sign of the reaction coordinate such that the reactant A is inx < 0.The kinetic equations for the relative populations of the reactantna(t) and productnc(t) are

na = −k+na + k−ncnc = +k+na − k−nc

(12.9)

In the equilibrium, the population ratio is given by the equilibrium constantK

nanc

=k−

k+= K (12.10)

The population decay is described by

na(t)− nana(0)− na

= e−λt (12.11)

whereλ = k+ + k−.We take the reaction coordinatex(q1, q2, · · · ) such that the reactant A is inx < 0.

Then,na = 〈θ(−x)〉 ≡ 〈θ−〉 (12.12)

The fluctuation ofδθ− ≡ θ− − 〈θ−〉 is

〈δθ2−〉 = 〈θ2

−〉 − 〈θ−〉2 = na − n2a = na(1− na) = nanc (12.13)

Now we consider the Onserger’s regression hypothesis

〈δθ−(0)δθ−(t)〉〈δθ2−〉

=na(t)− nana(0)− na

= e−λt (12.14)


which applies in the intermediate time scale after the initial transient timeτs and beforethe reactive time scale,

τe ≡1λ τ τs (12.15)

The time derivative of the left-hand-side is

〈δθ−(0)δθ−(t)〉〈δθ2−〉

= −〈δθ−(0)x(t)δ[−x(t)]〉〈δθ2−〉

= −〈δθ−(−t)x(0)δ[−x(0)]〉〈δθ2−〉

= −〈θ−(−t)x(0)δ[x(0)]〉〈δθ2−〉

(12.16)The first equality is explained as follows

δθ−(t) = θ−(t) =d

dtθ[−x(t)] = −x(t)δ[−x(t)] (12.17)

and the second equal is due to the steadyness of the equilibrium TCF, and the thirdequality comes from the fact that〈x(0)δ[−x(0)]〉 = 0, or that the average of the oddfunction vanishes, and thus the constant term inδθ− vanishes.δ(−x) = δ(x) is alsoused. The derivative of the right-hand-side is−λe−λt ∼ −λ for the intermediate timescalet ∼ τ under consideration. We thus get

〈θ−(−t)x(0)δ[x(0)]〉〈δθ2−〉

= λ = k+ + k− (12.18)

Combining this withk−/k+ = na/nc and using〈δθ2−〉 = nanc andna + nc = 1, we

get〈θ−(−t)x(0)δ[x(0)]〉

nanc= k+ + k− = k+

(1 +

nanc

)=k+

nc(12.19)

Finally, by notingna = 〈θ(−x)〉, we arrive at

k+(τ) =〈θ[−x(−τ)]x(0)δ[x(0)]〉

〈θ(−x)〉(12.20)

Note on the sign convention :

〈δθ−(0)δθ−(t)〉 = 〈1− θ[x(0)]− 〈θ−〉1− θ[x(t)]− 〈θ−〉〉

= (1− 〈θ−〉)2 − 2(1− 〈θ−〉)〈θ[x(0)]〉+ 〈θ[x(t)]〉

+〈θ[x(0)]θ[x(t)]〉

(12.21)

Therefore, using the relations already noted above, its time derivative is

〈θ[x(0)]θ[x(t)]〉 = 〈θ[x(0)]x(t)δ[x(t)]〉 = 〈θ[x(−t)]x(0)δ[x(0)]〉 (12.22)

12.7. κ(T ) FROM SIMULATION 77

The time dependent rate is thus

k+(τ) = −〈θ[x(−τ)]x(0)δ[x(0)]〉〈θ(−x)〉

(12.23)

Following this convention, we get

κ(t) = −〈δ[x(0)]x(0)θ[x(−t)]〉〈δ[x(0)]x(0)θ[x(0)]〉

= −〈j(0)θ[x(−t)]〉〈j+(0)〉

= −〈δ[x(0)]x(0)θ(+x) + θ(−x)θ[x(−t)]〉〈δ[x(0)]x(0)θ[x(0)]〉

= −〈j+(0)θ[x(−t)]〉〈j+(0)〉

− 〈j−(0)θ[x(−t)]〉〈j+(0)〉

= −〈j+(0)θ[x(−t)]〉〈j+(0)〉

+〈j−(0)θ[x(−t)]〉〈j−(0)〉

= −〈θ[x(−t)]〉+ + 〈θ[x(−t)]〉−(12.24)

Using the time-reversibility of the classical mechanics, the trajectories with the initialvelocity x(0) = v > 0 ending up in the product region at timet, x(t) = x > 0 isequivalent to the trajectory withx(0) = −v < 0 that was in the product region at time−t, x = x(−t). Therefore,

〈θ[x(t)]〉+ = 〈θ[x(−t)]〉− (12.25)

Note that this is irrelevant with the symmetry of the potential. Using this, we get

κ(t) = 〈θ[x(t)]〉+ − 〈θ[x(t)]〉− (12.26)

12.7.2 An alternative view

We shift the time in the TCF as

k+(t) =〈θ[−x(−t)]x(0)δ[x(0)]〉

〈θ(−x)〉=〈θ[−x(0)]x(t)δ[x(t)]〉

〈θ(−x)〉(12.27)

and use

δ[x(t)] =n∑k=1

1|x(tk)|

δ(t− tk) (12.28)

in whichtk is the set of times that givesx(tk) = 0. Then the numerator becomes

〈θ[−x(0)]x(t)δ[x(t)]〉 =n∑k=1

〈θ[−x(0)]x(t)1

|x(tk)|δ(t− tk)〉

=n∑k=1

〈θ[−x(0)]sign[x(tk)]δ(t− tk)〉(12.29)


The rate is then given by

k+(t) =n∑k=1

〈θ[−x(0)]sign[x(tk)]δ(t− tk)〉〈θ(−x)〉

(12.30)

The integration gives∫ τ

0

k+(t)dt =n∑k=1

〈θ[−x(0)]sign[x(tk)]〉〈θ(−x)〉

(12.31)

The meaning is clear. We start the trajectory from the equilibrium distribution in thereactant region and count the sign of the velocity when it passes the dividing surfaceat x(tk) = 0. The ensemble average of this gives the time integral of the ratek+(t).Of course, this formula is not very convenient for trajectory simulations with a highbarrier.

Chapter 13

Quantum Simulations

13.1 Wave Packet Dynamics

13.2 Gaussian Wave Packet

13.3 Time-dependent Variational Principle

13.4 Semiquantal Dynamics

13.5 Semiquantal Dynamics

13.5.1 1-Dim case

Extended Hamiltonian

Hext =p2

2m+

Π2

2m+ V0(x) +

h2

8mρ2+∑k

ρ2k

k!2kV

(2k)0 (x)

= up2

2m+

Π2

2m+ V0(x) +

h2

8mρ2+

12V

(2)0 (x)ρ2 +

18V

(4)0 (x)ρ4 + · · ·

(13.1)

79

80 CHAPTER 13. QUANTUM SIMULATIONS

EOM

x =∂Hext

∂p=

p

m; p = −∂He x t

∂x= −V ′0(x)−

∑kρ2k

k!2kV

(2k+1)0 (x)

= −V ′0(x)− ρ2

2V

(3)0 (x)− ρ4

8V

(5)0 (x) + · · ·

ρ =∂Hext

∂Π=

Πm

; Π = −∂He x t

∂ρ= +

h2

4mρ3−∑k

2kρ2k−1

k!2kV

(2k)0 (x)

= +h2

4mρ3− V (2)

0 (x)ρ− V(4)0 (x)

2ρ3 + · · ·

(13.2)

13.5.2 2-Dim case

V (x, y) = V (x, y) + Vx∆x+ Vy∆y +12!Vxx∆x2 +

12!Vyy∆y2 + Vxy∆x∆y

+13!Vxxx∆x3 +

13!Vyyy∆y3 +

12!Vxxy∆x2∆y +

12!Vxyy∆x∆y2

+14!Vxxxx∆x4 +

14!Vyyyy∆y4 +

13!Vxxxy∆x3∆y +

13!Vxyyy∆x∆y3

+1

2!2!Vxxyy∆x2∆y2

(13.3)Using〈∆x〉 = 〈∆y〉 = 0 and the decompositions

〈ABC〉 = 〈AB〉〈C〉+ 〈BC〉〈A〉+ 〈AC〉〈B〉 (13.4)

〈ABCD〉 = 〈AB〉〈CD〉+ 〈AC〉〈BD〉+ 〈AD〉〈BC〉 (13.5)

〈V (x, y)〉 = V (x, y) +12!Vxx〈∆x2〉+

12!Vyy〈∆y2〉+ Vxy〈∆x∆y〉

+34!Vxxxx〈∆x2〉2 +

34!Vyyyy〈∆y2〉2

+33!Vxxxy〈∆x2〉〈∆x∆y〉+

13!Vxyyy〈∆x∆y〉〈∆y2〉

+1

2!2!Vxxyy〈∆x2〉〈∆y2〉

(13.6)

Assuming non-correlated (diagonal), i.e.,〈∆x∆y〉 = 〈∆x〉 · 〈∆y〉 = 0and thus a factorized (Gaussian) packets :Ψ(x, y) = Ψx(x)Ψy(y)

Hext =p2x

2mx+

p2y

2my+

Π2x

2mx+

Π2y

2my+ V (x, y) +

h2

8mxρ2x

+h2

8myρ2y

+12Vxxρ

2x +

12Vyyρ

2y +

34!Vxxxxρ

4x +

12!2!

Vxxyyρ2xρ

2y +

34!Vyyyyρ

2y + · · ·

(13.7)

13.6 Path-integrals

82 CHAPTER 14. QUANTUM LIOUVILLE EQUATION

14.1.1 Time evolution

: (Consider the pure state)

∂

∂tρ(t) =

∂

∂t(|ψ(t)〉〈ψ(t)|) =

(∂

∂t|ψ(t)〉

)〈ψ(t)|+ |ψ(t)〉

(∂

∂t〈ψ(t)|

)Time-dependent Schrodinger Eq. (& Hermite conjugate)

∂

∂t|ψ(t)〉 = − i

hH|ψ(t)〉 &

∂

∂t〈ψ(t)| = +

i

h〈ψ(t)|H

⇒ Quantum Liouville Eq∂ρ

∂t= − i

h[H, ρ]

Mixed stateρ(t) = linear combination of the pure state⇒ extension of the above derivation involves only linear operations.⇒ the same quantum Liouville Eq applies for the mixed state

14.1.2 Example : 2-level system

H =[Haa Hab

Hba Hbb

]=[εa VabVba εb

], ρ(t) =

[ρaa(t) ρab(t)ρba(t) ρbb(t)

]∂

∂tρ(t) =

− ih

[Vabρba − Vbaρab (εa − εb)ρab + Vab(ρbb − ρaa)

(εb − εa)ρba + Vba(ρaa − ρbb) Vbaρab − Vabρba

]• Population (diagonal)⇐ Coherence (off-diagonal)

• Coherence⇐ (population difference) +∆ε × (self)

Can be rewritten as :

∂

∂t

ρaaρbbρabρba

= − ih

0 0 −Vba Vab0 0 Vba −Vab−Vab Vab εa − εb 0Vba −Vba 0 εb − εa

ρaaρbbρabρba

14.1.3 Liouville operator

(tetradic matrix, super-operator)

∂ρ

∂t= − i

h[H, ρ]≡ − i

hLρ︸︷︷︸

Matrix form of ρ ⇐ L is specified by four indices

∂∂tρmn = − i

h[(Hρ)mn − (ρH)mn] = − i

h

∑j

(Hmjρjn − ρmjHjn)

≡ − ih

∑j,k

Lmn,jkρjk

14.2. REDUCED DENSITY OPERATOR 83

Lmn,jk ≡ Hnjδnk − δmjHkn

14.2 Reduced density operator

• System + Bath :H = Hs(qs) +HB(QB) + V (qs,QB)

• Assume :Hs|ψi〉 = Ei|ψi〉 ,HB |χa〉 = εa|χa〉

• Abbreviate :|ψi(qs)〉 = |i〉 , |χa(Qa)〉 = |a〉

• Direct product|ia〉 = |i〉|a〉 (Completeness∑i,a |ia〉〈ia| = 1)

Note : |ia〉 are not eigenfunctions ofH because ofV

(though can be used for Tr calcs)

Expectation value :〈A(qs,QB)〉 = Tr[ρ(t)A(qs,QB)]

=∑i,a

〈ia|ρ(t)A|ia〉 =∑i,a

∑j,b

〈ia|ρ(t)|jb〉〈jb|A|ia〉

If looking at a “system” quantity (depends only onqs )

〈jb|A(qs)|ia〉 = 〈j|A(qs)|i〉〈b|a〉 = δab〈j|A(qs)|i〉

⇒ 〈A(qs)〉 =∑i,j

∑a

〈ia|ρ(t)|ja〉︸︷︷︸〈j|A(qs)|i〉

Reduceddensity operator :σ(t) ≡ TrB︸︷︷︸ ρ(t) =∑a

〈a|ρ(t)|a〉

σij(t) = 〈i|TrBρ(t)|j〉 =∑a

〈ia|ρ(t)|ja〉

⇒ 〈A(qs)〉 =∑i,j

σij(t)A(qs)ji = Trs[σ(t)A(qs)]

Note : TrB possesses the properties of projection⇒ Projection partition method on the Liouville eq ofρ(t)⇒ Reduced eq of motionfor σ(t)

14.3 Wigner Transformation

14.4 Mixed Quantum-classical Simulations

84 CHAPTER 14. QUANTUM LIOUVILLE EQUATION

Outlook

85

elements of molecular simulation methods

Documents