Dynamical, time-dependent view of molecular

theory

Yngve Ohrn and Erik Deumens

Quantum Theory Project Departments of Chemistry and Physics,

University of Florida PO BOX 118435, Gainesville, FL, 32611-8435

April 4, 2005

Abstract

In this paper we present a time-dependent, direct, nonadiabatic theoryof molecular processes. We put this approach in contrast to the currenttheory paradigm of approximate separation of electronic and nuclear dy-namics, which proceeds via approximate electronic stationary states andcorresponding potential energy surfaces. This established approach inall its variants has provided a basis for qualitative understanding of rateprocesses and, for systems with few nuclear degrees of freedom, it has pro-duced quantitative data that can be used to guide experiments. In thispicture the dynamics of the reacting system takes place on a stationaryelectronic state potential surface and may under the influence of nonadia-batic coupling terms “jump” to another potential surface the probabilityof such transitions often viewed as a statistical “surface hopping” [ J.Chem. Phys., 55 (1971) 562] event. The time-dependent, direct, andnonadiabatic theory presented here is fully dynamical in that the evolv-ing state, which describes the simultaneous dynamics of electrons andnuclei of a reacting system [Rev. Mod. Phys., 66(3), (1994) 917] changesin time under the mutual instantaneous forces in a Cartesian laboratorysystem of coordinates. This approach, which has been applied to reac-tive collisions involving polyatomic molecules over a large range of ener-gies, proceeds without predetermined potential energy surfaces, uses totalmolecular wave functions that are parameterized as generalized coherentstates, and imposes no constraints on molecular geometries.

1

1 Introduction

Chemistry may be characterized as the science of transformation of matter atlow to moderate energies. The study of chemical reactions, i.e. how prod-ucts are formed from reactants, is central to chemistry [1]. Since the beginningof modern chemistry one has studied kinetics and attempted to infer reactionmechanisms by varying controllable parameters, such as concentrations, temper-ature, substituents, etc. and determined the effect on product yields. However,not until recent times has the actual microscopic action of the reacting speciesbeen accessible to the chemist and the early theoretical models developed todescribe reactions were hence of a phenomenological nature. For instance, theearly work of Lindemann, Hinshelwood, Rice, Ramsparger, and Kassel on uni-molecular rate theory makes reasonable assumptions about the details of thedynamics of the reacting molecular system without actually knowing whetherthey are true or not. Later the seminal work of Eyring (J. C. Giddings andH. Eyring, J. Chem. Phys. 22 (1954) 538) introducing absolute reaction ratetheory, also known as activated complex theory or transition state theory, canbe labeled as the beginning of modern theoretical studies of chemical reactionrates by providing interpretations of the Arrhenius rate parameters in termsof molecular structure and properties. Common to this early theory and tomost current work on chemical reactions is the notion that the atomic nuclei ofparticipating species move subject to forces derived from a potential.

Chemical reactions in bulk are analyzed in terms of simple steps, calledelementary chemical reactions. Such elementary steps can be characterized asencounters of reactant molecules to form product molecules. Such encounterstake place in various media such as a solvent or at a metal surface. The perhapssimplest and purest form of elementary reactions take place in gas phase allowingexperimental control over initial and final states of participating species withthe use of modern laser technology, and thus obtaining detailed informationfrom which to draw conclusions about reaction mechanisms. This is often donein intricate molecular beam experiments involving ultrafast pulsed lasers tomonitor the reacting molecular system on picosecond to femtosecond time scales.In this way the experiment is not limited to viewing only the “opening act”, thereactants, and the “final act” the separated products, but can actually enjoythe entire “play” as the reaction proceeds from reactants to products.

The importance of the time parameter in the study of reactions is clearalready in the early kinetic studies, where it takes the form of the inverse rateconstant. From the point of view of fundamental chemical theory elementarychemical reactions are simply the detailed dynamics of electrons and atomicnuclei that constitute the the total molecular system of reacting species, whichis governed by the time-dependent Schrodinger equation

HΨ = i~∂Ψ

∂t. (1)

It is now a generally accepted view that electrons and atomic nuclei are “thefundamental particles” of chemistry and that the time-dependent Schrodinger

2

equation is the central equation for the study of molecular structure and dynam-ics, and thus also for chemical reactions in general. In spite of the tremendousadvances in the power and speed of electronic computers and in generally avail-able sophisticated software for finding adequate approximate solutions to theSchrodinger equation, accurate treatments are still limited to rather simple sys-tems, for which predictive results can be obtained from theory alone. Neverthe-less much can be achieved by approximations to the time-dependent Schrodingerequation and even with approximate solutions to approximate equations.

For time-independent Hamiltonians H one often equivalently studies thetime-independent Schrodinger equation

HΦ = EΦ (2)

not as a boundary value problem but seeking solutions for general energy valuesE. The explicitly time-dependent formulation is more readily treated as aninitial value problem.

The disparate masses of even the lightest atomic nucleus and an electronhave led to the widely accepted view that an adequate description of low energyprocesses can be achieved by assuming an effective separation of electronic andnuclear degrees of freedom. Considering the electron dynamics to take place inthe field of stationary nuclei leads to the introduction of an electronic Hamil-tonian operator Hel consisting of the kinetic energy operator of electrons, theirmutual Coulombic interactions, and the Coulombic attraction terms to each ofthe atomic nuclei. The corresponding Schrodinger eigenvalue problem

Hel(~R)|n〉 = En(~R)|n〉 (3)

has the solutions |n〉, which are electronic stationary states with characteristic

electronic energies En(~R). The lowest eigenvalue E0(~R) corresponds to theelectronic ground state |0〉 can be obtained at some level of approximation for

various chosen nuclear geometries ~R in an internal coordinate system attachedto the nuclear framework. For N nuclei such a function in 3N − 6 dimensionsis commonly interpolated and fitted to some analytical representation to alsoyield the energy at intermediate nuclear geometries. As a practical matter mostelectronic structure codes use Cartesian coordinates to solve Eq.(3). Addingthe Coulomb repulsion terms of the nuclei to this function one obtains theground state potential energy surface (PES). Commonly this PES becomes thepotential energy for the nuclear dynamics, which can be treated classically,semi-classically, or fully quantum mechanically.

Obviously, the electronic energies En(~R) for n 6= 0 corresponds in a similarmanner to potential surfaces for electronically excited states. Each PES usuallyexhibits considerable structure for a polyatomic system and will provide usefulpictures with reactant and product valleys, local minima corresponding to stablespecies, and transition states serving as gateways for the system to travel fromone valley to another. However for the number of nuclear degrees of freedombeyond six, i.e. for more than four atom systems it becomes extremely cum-bersome to produce the PES’s and quite complicated to visualize the topology.

3

Furthermore, when more than one PES is needed, which is not unusual, thereis a need for nonadiabatic coupling terms, which also may need interpolation inorder to provide useful information.

For those few systems for which one or more accurate PES have been de-termined this strategy, of proceeding via precalculated potentials, works quitewell [2]. Detailed quantum dynamics obtains accurate differential and integralstate-state cross sections and rate coefficients in agreement with the best ex-periments for some small systems. However, as the complexity of the reactingsystem increases it becomes increasingly difficult to proceed via full potentialenergy surfaces. One way out of this problem is to identify some active modesand eliminate or discretize degrees of freedom that are either changing slowlythroughout the critical part of the dynamics or which are not directly involved.This reduced dimensionality dynamics [3] has been successful in some cases, butintroduces some arbitrariness or bias and hard to control errors. Another pro-cedure that has gained recent prominence is so called direct classical dynamics(see e.g. J. B. Liu, K. Y. Song, and W. L. Hase, JACS 126 (2004) 8602). Thisapproach calculates the forces on the nuclei during a classical trajectory mak-ing it necessary to compute the PES only in those points where the dynamicstake the nuclei. Since the reacting system can tumble one commonly performsthe calculations in 3N − 3 dimensions also including the rotational degrees offreedom but no coupling terms.

In the following sections we consider the Coulombic Hamiltonian of a generalmolecular system and comment upon the difference between using an internal setof coordinates with axes fixed in the molecular system and employing a Carte-sian laboratory system of axes. The study of reactive molecular systems in termsof stationary molecular electronic states and their potential energy surfaces asdescribed above is put in contrast to a direct, nonadiabatic, time-dependenttreatment, which is fully dynamical in that the evolving state, which describesthe simultaneous dynamics of participating electrons and nuclei, changes in timeunder the mutual instantaneous forces. The wave function parameters carry thetime-dependence and in the choice of parameters it is useful to consider general-ized coherent states. This approach proceeds without predetermined potentialenergy surfaces, and the dynamical equations that describe the time evolutionof the total system state vector are derived using the time-dependent variationalprinciple. A minimal form of this dynamical approach to molecular processes isdiscussed in some detail.

2 Molecular Hamiltonian

The molecular Hamiltonian contains a variety of terms. If we limit the descrip-tion to Coulombic interactions we can write (we are using the subscripts i andj for electron labels and k and l for nuclear labels)

4

H = − ~2

2m

∑

i

∇2xi

− ~2

2

∑

k

1

mk∇2

Xk−∑

i

∑

k

Zke2

|~xi − ~Xk|

+1

2

∑

i

∑

j 6=i

e2

|~xi − ~xj |+

1

2

∑

k

∑

l

ZkZle2

| ~Xk − ~Xl|, (4)

where the terms in order are the operators of kinetic energy of the electrons,kinetic energy of the nuclei, the electron-nuclear attraction energy, the electron-electron repulsion energy, and the nuclear-nuclear repulsion energy. Obviouslyif it were not for the electron-nuclear attraction terms the electronic and thenuclear energetics and dynamics would be decoupled. Although this term is notsmall much of molecular quantum mechanics can be performed with an effectivedecoupling of electronic and nuclear degrees of freedom or rather a discretizationof the nuclear coordinates.

One normally proceeds by eliminating the translational motion and choosesthe origin of a molecule-fixed coordinate system. A suitable choice is the centerof mass of the nuclei. This choice introduces no additional coupling terms be-tween the nuclear and the electronic degrees of freedom. However, it introducesreduced masses and so called mass polarization terms. Such terms are of theform

~2

2M0

∑

i

∑

j 6=i

∇i∇j (5)

for the electrons and a similar term for the nuclei, where M0 is the total massof all the nuclei. Because of this small factor these terms are small and oftenneglected. Nevertheless, these omissions must be recognized as a source oferrors. The potential energy terms (collectively given the symbol U) all dependon the interparticle distances, which are unaffected by the transformation tointernal coordinates. Obviously, the choice of an internal origin of coordinatesleads to the elimination of three degrees of freedom, which means, say, that theposition of one of the nuclei is dependent on the position of all the others.

• Example We consider a general molecular system in the laboratory framewith the center of mass

~ρ =1

M

[

∑

k

mk~Xk +m

∑

i

~xi

]

. (6)

The internal coordinates relative to the center of mass of the nuclei are

~ri = ~xi −1

M0

∑

k

mk~Xk (7)

for the electrons, and

5

~Rl = ~Xl −1

M0

∑

k

mk~Xk (8)

for the nuclei. The position of one nucleus, say p is then obtained as

~Rp = − 1

mp

∑

k 6=p

mk~Rk (9)

from the fact that the center of mass of the nuclei is the origin. In the aboveexpressions we have used the notations M0 =

∑

k mk and M = M0 +mN ,where N is the number of electrons in the system.

The kinetic energy terms are now altered and we can see how they changeby using the chain rule of differentiation. Note that the Cartesian compo-nents of the position coordinates are such that

~ρ = (ξ, η, ζ),

~x = (a, b, c)

~X = (A,B.C),

~r = (x, y, z),

~R = (X,Y, Z).

and for example

∂

∂a=

∂

∂x

∂x

∂a+

∂

∂ξ

∂ξ

∂a=

∂

∂x+m

M

∂

∂ξ(10)

yielding

∇xi= ∇i +

m

M∇ρ. (11)

Similarly we obtain for k 6= p

∂

∂Ak=

∂

∂Xk

∂Xk

∂Ak+

∂

∂ξ

∂ξ

∂Ak+∑

l

∂

∂Xl

∂Xl

∂Ak+∑

i

∂

∂xi

∂xi

∂Ak

=∂

∂Xk+mk

M

∂

∂ξ− mk

M0

∑

l6=p

∂

∂Xl− mk

M0

∑

i

∂

∂xi

leading to

∇Xk= ∇k +

mk

M∇ρ − mk

M0

∑

l6=p

∇l −mk

M0

∑

i

∇i (12)

6

(for k 6= p) and for the particular nucleus p

∇Xp=mp

M∇ρ − mp

M0

∑

l6=p

∇l −mp

M0

∑

i

∇i (13)

Insertion of these expressions in the molecular Hamiltonian yields [4]

H(p) = − ~2

2M∇2

ρ − ~2

2

[

1

m+

1

M0

]

∑

i

∇2i −

~2

2M0

∑

i6=j

∇i∇j

−~2

2

∑

k 6=p

[

1

mk− 1

M0

]

∇2k +

~2

2M0

∑

k 6=lk,l6=p

∇k∇l + U (14)

In the above example the choice of molecule fixed origin is the center ofmass of the nuclei, but any other point could have been chosen, and a differentchoice will give a different internal Hamiltonian. For instance, if we choose theorigin to be centered on a particular nucleus p , which might be the preferredchoice if that nucleus has a much greater mass than all the others, then thecorresponding Hamiltonian is

H(p) = − ~2

2M∇2

ρ − ~2

2

[

1

m+

1

mp

]

∑

i

∇2i −

~2

2mp

∑

i6=j

∇i∇j

− ~2

mp

∑

k 6=p

∑

i

∇k∇i −~

2

2

∑

k 6=p

[

1

mk+

1

mp

]

∇2k

− ~2

2mp

∑

k 6=lk,l6=p

∇k∇l + U, (15)

where coupling terms between the nuclear and the electronic degrees of freedomappears in the kinetic energy.

In general one can use a product wave function with one factor dependingonly on the center of mass coordinates ~ρ and a second factor depending on therest, i.e. all the internal coordinates and the three rotational degrees of freedom.

One normally proceeds by defining the electronic Hamiltonian

Hel = −~2

2

[

1

m+

1

M0

]

∑

i

∇2i −

~2

2M0

∑

i6=j

∇i∇j −∑

i

∑

k

Zke2

|~ri − ~Rk|

+1

2

∑

i

∑

j 6=i

e2

|~ri − ~rj |+

1

2

∑

k

∑

l

ZkZle2

|~Rk − ~Rl|, (16)

7

where the nuclear coordinates are now representing a fixed nuclear framework(classical nuclei held fixed). The solution of the eigenvalue problem of theelectronic Schrodinger equation

HelΦk(~r; ~R) = Vk(~R)Φk(~r; ~R) (17)

for a fixed nuclear geometry yields the (approximate) electronic stationary statesand the associated energy eigenvalues for that single nuclear configuration. Thisis then commonly repeated for a number of different nuclear geometries usuallychosen by some rationale so as to cover a particular minimum in the energy orsome barrier , etc. If a sufficient number of nuclear geometry points are usedand they are chosen to cover all possible distortions of the molecule we get oneor more potential energy surfaces Vk(~R). The (approximate) eigenfunctions andthe associated potential energy surfaces are known only in a number of discretepoints in 3N − 6 dimensions (for N nuclei), since for their determination also

the rotational motion is assumed to have been eliminated (Vk(~R) is rotationinvariant) and the electronic stationary states and potential energy surfacesare constructed in a molecule-fixed coordinate system. Further use of thesequantities for the study of molecular processes involving nuclear motion mustthen involve interpolation of some form. Note that the common procedure ofjust neglecting the nuclear kinetic energy terms in the total Hamiltonian Eq.(4) will not produce the reduced masses and the mass polarization terms.

The Schrodinger equation for the system when the center of mass kineticenergy term has been eliminated can be expressed as

[Tn +Hel]Ψ(~r, ~R) = EΨ(~r, ~R), (18)

where

Tn = −~2

2

∑

k 6=p

[

1

mk− 1

M0

]

∇2k +

~2

2M0

∑

k 6=lk,l6=p

∇k∇l. (19)

It is then common to introduce the stationary electronic states as a basis,such that

Ψ(~r, ~R) =∑

k

Φk(~r; ~R)χk(~R) (20)

and by inserting this into Eq.(18) , multiplying from the left by Φ∗l , integrating

over the electronic degrees of freedom, and using the fact that the stationaryelectronic states at a fixed nuclear geometry are orthogonal, one obtains

[(Φl | TnΦl)r + Vl(~R)]χl(~R) −Eχl(~R)

= −∑

k 6=l

(Φl | TnΦk)rχk(~R). (21)

8

This is a set of coupled partial differential equations with the terms (Φl|TnΦk)r

on the right hand side being the so called nonadiabatic coupling terms, with thesubscript r indicating integration over electronic coordinates. These equationsform the basis of the close-coupling approach to atomic and molecular scatter-ing [5]. When these terms are neglected we obtain what is called the adiabaticapproximation. The nuclear dynamics is then described by the Schrodingerequation

[(Φl|TnΦl)r + Vl(~R)]χl(~R) = Eχl(~R). (22)

The term (Φl|TnΦl)r can be expressed in more detail using the fact that

−~2

2

∑

k 6=p

[

1

mk− 1

M0

]

(Φl | ∇2kΦl)r +

~2

2M0

∑

k 6=mk,m6=p

(Φl | ∇k∇mΦl)r (23)

is an operator that acts on the nuclear wave function χl(~R), and, for instance,

(Φl|∇2kΦl)rχl(~R) = ∇2

kχl(~R) + 2(Φl|∇kΦl)r · ∇kχl(~R)

+χl(~R)(Φl|∇2kΦl)r. (24)

Now, the integral (Φl|∇kΦl)r , over electronic degrees of freedom involves dif-ferentiation with respect to parameters under the integral sign and since theelectronic stationary states are assumed to be normalized and orthogonal, weget for the case of real wave functions

(Φl|∇kΦl)r =1

2∇k(Φl|Φl)r = 0. (25)

So, we can define the potential energy term

Bl(~R) = −~2

2

∑

k

(

1

mk− 1

M0

)

(Φl|∇2kΦl)r

+~

2

2M0

∑

k 6=mk,m6=p

(Φl|∇k∇mΦl)r, (26)

which is usually small but not necessarily unimportant. The Schrodinger equa-tion for the nuclear motion then becomes

[Tn + Vl(~R) +Bl(~R)]χl(~R) = Eχl(~R), (27)

and when the mass polarization terms in the kinetic energy operator are ne-glected we can write

9

−~2

2

∑

k 6=p

(

1

mk− 1

M0

)

∇2k + Vl(~R) +Bl(~R)

χl(~R) = Eχl(~R), (28)

which is the normal result of the so called adiabatic approximation. When alsothe Bl(~R), is neglected one calls the result the Born-Oppenheimer approxima-tion (M. Born and J. R. Oppenheimer, Ann. Physik 84 (1927) 457).

• Example We study the case of a diatomic molecule. The kinetic energyoperator can for this case be expressed as

−~2

2

(

1

m1− 1

M0

)

∇21, (29)

and when ~R = ~R1− ~R2, and the center of mass of the nuclei is the origin, so~R2 = −m1

m2

~R1,which yields ~R = (m1 +m2)~R1/m2, and the kinetic energy

can be expressed in terms of ~R as

−~2

2

(

1

m1− 1

M0

)

∇21 = −~

2

2

(

1

m1+

1

m2

)

∇2 = − ~2

2µ∇2. (30)

The above equation then becomes

[

− ~2

2µ∇2 + Vl(R) +Bl(R)

]

χl(~R) = Eχl(~R) (31)

where

− ~2

2µ∇2 = − ~

2

2µ

[

∂2

∂R2+

2

R

∂

∂R

]

+~J2

2µR2, (32)

with ~J2, the total angular momentum operator square of rotational motionand µ = m1m2/(m1 + m2). Since for a diatomic the potential energydepends only on the bond distance R, the wave functions can be expressedas

χ(~R) = υ(R)YJM (θ, ϕ), (33)

where YJM (θ, ϕ), is a spherical harmonic and the Schrodinger equationfor vibrational motion is

[

− ~2

2µ

(

∂2

∂R2+

2

R

∂

∂R

)

+~

2J(J + 1)

2µR2+ V (R) +B(R)

]

υ(R)

= Eυ(R), (34)

showing the so called centrifugal term. This illustrates that the vibrationaland rotational motions of a molecule are indeed coupled.

10

3 The Time-Dependent Variational Principle inQuantum Mechanics

The time-dependent variational principle in quantum mechanics [6] starts fromthe quantum mechanical action [7, 8]

A =

∫ t2

t1

L(ψ∗, ψ)dt, (35)

where the quantum mechanical Lagrangian is

L(ψ∗, ψ) =

⟨

ψ|i~ ∂∂t

−H |ψ⟩

/ 〈ψ|ψ〉 , (36)

and H is the quantum mechanical Hamiltonian of the system. When the wavefunction ψ is completely general and allowed to vary in the entire Hilbert spacethen the TDVP yields the time-dependent Schrodinger equation. However, if thepossible wave function variations are restricted in any way, such as is the case fora wave function represented in a finite basis and being of a particular functionalform, then the corresponding Lagrangian will generate an approximation to theSchrodinger time-evolution.

We consider a wave function expressed in terms of a set of (in general com-plex) parameters z (e.g. molecular orbital coefficients, average nuclear positionsand momenta, etc.). These parameters are time-dependent and can be expressedas zα ≡ zα(t) and thought of as arranged in a column or row array. We write

ψ = ψ(z) = |z〉 (37)

and employ the principle of least action

δA =

∫ t2

t1

δL(ψ∗, ψ)dt = 0 (38)

with the Lagrangian

L =

[

i

2〈z|z〉 − i

2〈z|z〉 − 〈z|H |z〉

]

/ 〈z|z〉 , (39)

where we have put ~ = 1, and write the symmetric form of the time derivativeterm. One way to see how this can come about is to consider

∫ t2

t1

∂∂t 〈z|z〉〈z|z〉 dt = 0 (40)

which holds if we require

〈z(t2)|z(t2)〉 = 〈z(t1)|z(t1)〉, (41)

as our boundary condition.The variation of the Lagrangian can be expressed in more detail as

11

δL =i

2[〈δz|z〉 − 〈δz|z〉 − 〈δz|H |z〉] / 〈z|z〉

−[

i

2〈z|z〉 − i

2〈z|z〉 − 〈z|H |z〉

]

× 〈δz|z〉 / 〈z|z〉2

+complex conjugate (42)

We would like to get rid of all the terms that contain the variation δz. Tothis end we add and subtract the total time derivative

d

dt

〈δz|z〉〈z|z〉 =

〈δz|z〉 + 〈δz|z〉〈z|z〉 − 〈δz|z〉

〈z|z〉2d

dt〈z|z〉 , (43)

and its complex conjugate to write

δL =i

2

〈δz|z〉〈z|z〉 +

i

2

〈δz|z〉〈z|z〉 − i

2

〈δz | z〉〈z|z〉2

d

dt〈z|z〉 − i

2

d

dt

〈δz|z〉〈z|z〉

−〈δz|H |z〉〈z|z〉

−〈δz|z〉〈z|z〉2

[

i

2〈z|z〉 − i

2〈z|z〉 − 〈z|H |z〉

]

+complex conjugate. (44)

The time integration involved in δA = 0 eliminates the total derivative termssince due to the boundary conditions they are zero, i.e.

〈δz(t2)|z(t2)〉 − 〈δz(t1)|z(t1)〉 = 0, (45)

which follows from Eq.(41) and the fact that |δz〉 and 〈δz| are independentvariations.

The surviving terms of δL can be expressed as

i〈δz|z〉〈z|z〉 − 〈δz|H |z〉

〈z|z〉 − 〈δz|z〉〈z|z〉2

[i 〈z|z〉 − 〈z | H | z〉]

+complex conjugate. (46)

Since δz and δz∗ can be considered as independent variations one can con-clude that

(

i∂

∂t−H

)

|z〉 =〈z|i∂/∂t−H |z〉

〈z|z〉 |z〉, (47)

which is the Schrodinger equation if the right hand side is zero. By explicitlyconsidering the overall wave function phase we can eliminate the right handside. We write

12

|z〉 −→ e−iγ |z〉 (48)

with γ only a function of time and obtain

〈z|i∂/∂t−H |z〉 −→⟨

z|eiγ(i∂/∂t−H)e−iγ |z⟩

= γ 〈z|z〉 + 〈z|i∂/∂t−H |z〉 = 0, (49)

which means that the time derivative of the overall phase must be

−γ =〈z|i∂/∂t−H |z〉

〈z|z〉 . (50)

We introduce the notations S(z, z∗) = 〈z|z〉, and E (z, z∗)= 〈z|H |z〉 / 〈z | z〉,which leads to the equation

−γ =i

2

∑

α

[

zα∂

∂zα− z∗α

∂

∂z∗α

]

lnS(z, z∗) −E (z, z∗) , (51)

where we have used the chain rule of differentiation. Note that for a stationarystate all z = 0, and E (z, z∗) = E yielding γ = Et and the phase factor e−iEt/~.The above expression for δA can be similarly written as

0 = δA =

∫ t2

t1

δLdt (52)

=

∫ t2

t1

{∑

α

[∑

β

i∂2 lnS

∂z∗α∂zβzβ − ∂E

∂z∗α]δz∗α (53)

+∑

α

[∑

β

−i ∂2 lnS

∂zα∂z∗βz∗β − ∂E

∂zα]δz∗α}dt, (54)

where, say δz =∑

α∂

∂zαδzα, and since δzαand δz∗α are independent variations

we can write

i∑

β

Cαβ zβ =∂E

∂z∗α, (55)

−i∑

β

C∗αβ z

∗β =

∂E

∂zα, (56)

where Cαβ = ∂2 lnS/∂z∗α∂zβ. We introduce the matrix C = {Cαβ} and assumeit to be invertible to write

[

z

z∗

]

=

[

−iC−1 0

0 iC∗−1

] [

∂E/∂z∗

∂E/∂z

]

, (57)

13

which is a matrix equation in block form. One may introduce a generalizedPoisson bracket by considering two general differentiable functions f(z, z∗), andg(z, z∗) and write

{f, g} =[

∂f/∂zT ∂f/z†]

[

−iC−1 0

0 iC∗−1

] [

∂g/∂z∗

∂g/∂z

]

= −i∑

α,β

[

∂f

∂zα

(

C−1)

αβ

∂g

∂z∗β− ∂g

∂zα

(

C−1)

αβ

∂f

∂z∗β

]

. (58)

It follows that

z = {z, E} ,z∗ = {z∗, E} , (59)

which shows that the time evolution of the wave function parameters, and thus,of the wave function, is governed by Hamilton-like equations. Such a set ofcoupled first-order differential equations in time plus the equation for the evo-lution of the overall phase can be integrated by a great variety of methods.Schematically we write z(t) = F (z(t)) and proceed by finite steps such that∆zi = F (zi) ∆t and zi+1 = zi + ∆zi; z(0) = z0.

4 Coherent States

The discussion of the TDVP in the previous chapter exploits a family of statevectors |z〉 labeled by a set of time-dependent, and complex parameters z ={z1, z2, · · · , zM} [9]. Such parameter spaces should be continuous and complete

in the sense that as the state vector evolves in time and the complex parametersassume all possible values throughout their range, all states of the particularform |z〉 are obtained. Such demands on parameter spaces are satisfied by(generalized) coherent states [10] which relates the parameters to a particularLie groupG. Typically one chooses a unitary irreducible representation of G anda corresponding lowest (or highest) weight state |0〉 of such a representation. Amaximal subgroup H of G that leaves |0〉 invariant is called the stability groupand the cosets of G by H provide a suitable non-redundant set of parametersto label the coherent state. In general one would also require the existence of apositive measure dz on this parameter space such that when the integral

∫

|z〉|〈z|dz = I (60)

is taken over the range of the parameter space one obtains the identity.Already the notion of continuity of the labels rules out as coherent states

some familiar sets of states used in quantum mechanics. For instance, a set ofdiscrete orthogonal states, such as a a set of orthonormal basis functions {|n〉}cannot be coherent states.

14

4.1 Gaussian wave packet as a coherent state

A Gaussian wave packet in one dimension can be expressed as (~ = 1)

ψ(x) ∝ exp

[

−1

2

(

x− q

b

)2

+ ipx

]

, (61)

where we take the view that the parameters p, and q are time dependent, whilethe width parameter b is time-independent. The interpretation of these param-eters are evident from the definition of quantum mechanical averages, i.e.

〈x〉 =

∫ ∞

−∞

xe−( x−q

b )2

dx/

∫ ∞

−∞

e−( x−q

b )2

dx = 〈ψ|xψ〉 / 〈ψ|ψ〉 . (62)

Since the average value 〈x− q〉 = 0, it follows that q = 〈x〉, which is the averageposition of the wave packet. The average momentum of the wave packet

〈p〉 =

⟨

−i ∂∂x

⟩

= p− 〈x− q〉 /b2 = p (63)

defines the parameter p. The square of the width of the wave packet

⟨

(x− q)2⟩

= −∂ 〈ψ|ψ〉 /∂(

1/b2)

/ 〈ψ|ψ〉 =√πb3

2/√πb (64)

making the width ∆x =⟨

(x− q)2⟩1/2

= b/√

2.

The Gaussian wave packet has a number of interesting properties. For in-stance, it has a minimal uncertainty product ∆x∆p = 1/2 in units of ~. Thisfollows from

〈p2〉 = 〈−~2 d

2

dx2〉 = p2 +

~2

2b2(65)

and ∆p =⟨

p2 − p2⟩1/2

= 1/b√

2, (~ = 1).In addition the Gaussian wave packet can be written as a displaced harmonic

oscillator ground state, such that

ψ(x) = exp [ipx] exp [−iqp] exp

[

−1

2

(x

b

)2]

, (66)

i.e. the oscillator ground state is displaced, x −→ x − q, and boosted 0 → p.This can be seen from

e−iqp exp

[

−1

2

(x

b

)2]

=

[

1 − q∂

∂x+q2

2

∂2

∂x2− ...

]

exp

[

−1

2

(x

b

)2]

= exp

[

−1

2

(

x− q

b

)2]

. (67)

15

The Gaussian wave packet is a coherent state and can be expressed as asuperposition of oscillator states. This means that

ψ(x) =∑

n

cn |n〉 = exp[

za† − z∗a]

|0〉 , (68)

where |n〉 is a harmonic oscillator eigenstate a and a† are harmonic oscilla-tor field operators and z is a suitable complex combination of wave functionparameters.

This can be seen from the result

ψ(x) ∝ eipxe−iqpe−1

2 (xb )

2

∝ e−i(qp−px)e−1

2 (xb )

2

, (69)

where, since x and p do not commute, the last step is nontrivial. Introducingthe complex parameter z = (q/b+ ibp) /

√2, and observing that the harmonic

oscillator field operators can be expressed as

a† = −i (bp+ ix/b) /√

2, (70)

a = i (bp− ix/b) /√

2, (71)

we can write za† − z∗a = −i (qp− px), and

ψ(x) ∝ eza†−z∗ae−1

2 (xb )

2

∝ eza†−z∗a |0〉 . (72)

The last expression is the ”classical” or canonical coherent state |z〉. TheBaker-Campbell-Hausdorff (BCH) formula, yields

eA+B = exp

[

−1

2[A,B]−

]

eAeB , (73)

which is true for the case when the commutator [A,B]− commutes with A andB. When applied to

|z〉 = eza†−z∗a |0〉 (74)

the BCH formula yields

|z〉 = e−1

2|z|2eza†

e−z∗a |0〉

= e−1

2|z|2eza† |0〉 = e−

1

2|z|2

∞∑

n=0

(n!)−1 (

za†)n |0〉

= e−1

2|z|2

∞∑

n=0

(n!)−1/2

(z)n |n〉 . (75)

The Gaussian wave packet in this form is the original ”coherent state”. Gen-eralizations of this concept have been made, in particular the work of Perelo-mov [11] has introduced so called group related coherent states. Such a state is

16

formed by the action of a Lie group operator exp {∑m zmFm} acting on a refer-ence state |0〉. The {zm} are the, in general complex, Lie group parameters, and{Fm} are the generators of the corresponding Lie algebra. The reference stateis usually a lowest weight state and called the fiducial state. It is commonlyinvariant to some of the group elements, thus defining a so called stability groupof the fiducial state. The parameters labeling the coherent state are then asso-ciated with the left coset of the Lie group with respect to the stability group.This assures non-redundency of parameters. The canonical coherent state hasthis form in terms of the so called Weyl group, whose Lie algebra generators are{

1, a, a†}

. The one parameter stability group is just the phase factor eiα and

the coset representative is eza†−z∗a.The scalar product of two coherent states

〈z1| z2〉 = exp{

−(

|z1|2 + |z2|2)

/2}

∑

n

(z∗1z2)n

n!

= exp

{

−1

2|z1|2 + z∗1z2 −

1

2|z2|2

}

, (76)

i.e. a nowhere vanishing continuous function of the parameters. The canonicalcoherent state has the property that

a |z〉 = z |z〉 , (77)

which can easily be seen from

e−za†

aeza†

= a+ z[

a, a†]

−+z2

2

[

[

a, a†]

−, a†]

−+ ... = a+ z, (78)

which means that e−za†

aeza† |0〉 = z |0〉. Furthermore, the coherent state for allthe values of the complex parameter z is a set of states satisfying the resolutionof the identity

π−1

∫

|z〉 〈z| d2z =1

π

∑

n,m

(n!m!)−1/2

∫

e−|z|2 (z∗)n

(z)m |m〉 〈n| d2z

=1

π

∑

n,m

(n!m!)−1/2∫ ∞

0

e−|z|2 |z|n+m+1 d |z|

×∫ 2π

0

ei(m−n)φdφ |m〉 〈n|

=∑

n

(n!)−1∫ ∞

0

e−|z|2 |z|2nd |z|2 |n〉 〈n|

=∑

n

|n〉 〈n| = 1. (79)

This result permits us to write

17

|z′〉 = π−1

∫

|z〉 〈z| z′〉d2z, (80)

which illustrates the overcompleteness of the set {|z〉}. Thus, as a set of func-tions labeled by the continuous complex parameter z the coherent state satisfiesthe resolution of the identity and is inherently linearly dependent.

Considering the time evolution of a harmonic oscillator with |z〉 as the initialstate, we obtain

e−iHt |z〉 = e−itω(n+ 1

2 ) |z〉

= e−1

2|z|2

∞∑

n=0

(n!)−1/2 (

ze−itω)n |n〉 e−itω/2 ∝

∣

∣e−itωz⟩

. (81)

This shows that the coherent state evolves into other coherent states by a time-dependent label change that follows the classical oscillator solution.

Application of the TDVP to the wave packet dynamics with the coherentstate |z〉 is straightforward. We note that

S(z∗, z′) = 〈z|z′〉 = exp

(

−1

2|z|2 + z∗z′ − 1

2|z′|2

)

, (82)

E(z∗, z) = 〈z|H |z〉 / 〈z|z〉 = ω 〈z|(

a†a+1

2

)

|z〉 / 〈z|z〉

= ω

(

z∗z +1

2

)

, (83)

and that the dynamical equations become

[

i 00 −i

] [

zz∗

]

=

[

ωzωz∗

]

, (84)

since C = ∂2 lnS/∂z∗∂z′ |z′=z= 1. The equation iz = ωz becomes in moredetail

i√2

(q/b+ ibp) = ω1√2

(q/b+ ibp) (85)

assuming a constant width wave packet. One easily deduces that

p = −ω2p, (86)

q = −ω2q, (87)

i.e. in an oscillator field with b = 1/√mω the Gaussian wave packet is coherent

and that its average position has a harmonic motion q(t) = q0 cosωt+ p0

mω sinωt,while p (t) = p0 cosωt−mω sinωt.

18

4.1.1 Gaussian wave packet with evolving width

A more general coherent state description of a Gaussian wave packet is requiredwhen we allow the width parameter to evolve in time. The corresponding Liegroup is then Sp(2, R), which is isomorphic to SU(1, 1) or SO(2, 1). The gen-erators of the sp(2, R) Lie algebra are

t1 =i

2

[

−1 00 1

]

, t2 =i

2

[

0 11 0

]

, t3 =i

2

[

0 1−1 0

]

, (88)

satisfying the relations

[t1, t2] = −it3, [t2, t3] = it1, [t3, t1] = it2, (89)

where the different signs on the right indicate that we are dealing with a non-compact group.

Realization of the generators in terms of a Cartesian coordinate x and itsconjugate momentum p, such that [x, p] = i~, are

t1 → T1 = −(xp+ px)/4~

t2 → T2 = (p2/2µ− µω2x2/2)/2~ω (90)

t3 → T3 = (p2/2µ+ µω2x2/2)/2~ω.

Another useful realization obtains in terms of the oscillator field operator

a =

(

ip/√

µω~ +

√

µω

~x

)

/√

2 (91)

and its adjoint. We write

T+ = −T2 + iT1 =1

2a†a†

T− = −T2 − iT1 =1

2aa (92)

T0 = T3 =1

4(a†a+ aa†).

A common parameterization of the Sp(2, R) group is in terms of Euler angles,such that a group element would be

g(α, β, γ) = eiαt3eiβt1eiγt3 . (93)

One can readily show that

eiβt1 =

[

eβ/2 0

0 eβ/2

]

(94)

and

19

eiγt3 =

[

cos γ/2 − sin γ/2sin γ/2 cos γ/2

]

. (95)

The operator exp [iβT1] is a scale transformation as can be seen from the relation

eiβT1f(x) = exp

[

−β2x∂

∂x− β

4

]

f(x) = e−β/4f(e−β/2x). (96)

This result is readily shown by considering the transformation of powers of thecoordinate. For instance, by using the power series representation

f(x) =

∞∑

n=0

f (n)(0)

n!xn (97)

and the defining expansion of an exponential operator

exp

[

−β2

(

x∂

∂x+

1

2

)]

=

∞∑

k=0

βk

k!

[

−β2

(

x∂

∂x+

1

2

)]k

, (98)

noting that

[

−1

2

(

x∂

∂x+

1

2

)]k

xn = (−1)k

(

2n+ 1

4

)k

xn (99)

and

[

∞∑

k=0

(−β4

)k1

k!

]

×[

[

∞∑

l=0

(−β2

)l1

l!x

]n

=

[

∞∑

k=0

(−β4

)k1

k!

]

×[

[

∞∑

l=0

(−nβ2

)l1

l!

]

xn

= xn∞∑

j=0

(−β4

)j1

j!

j∑

k=0

(

j

k

)

(2n)j−k

=

∞∑

j=0

(−(2n+ 1)β

4

)jxn

j!(100)

the scaling property is shown.As discussed above, defining a coherent state involves choosing a Lie group

G, a unitary irreducible representation, the lowest (or highest) weight state ofsuch a representation, a stability subgroup, and corresponding cosets. Thiswill yield a useful parameter space in which to describe the coherent state. Inanalogy with the compact Lie group SO(3), the irreducible representations ofSp(2, R) are labeled by the eigenvalues of T3 corresponding to the lowest weightstate. The harmonic oscillator ground state

20

|0〉 ∝ exp

[

−1

2

(x

b

)2]

(101)

with b2 = ~/µω is the lowest weight state of an irreducible representation labeledby k = 1/8, where

T3|0〉 = 2k|0〉. (102)

We then identify the stability group H as the set

H = h|T (h)|0〉 = eiσh |0〉. (103)

Each element of the coset space G/H then corresponds to a coherent state. Thedecomposition of the group into cosets, taking advantage of the stability groupproperties, reduces the parameter space of the coherent state to a non-redundantset. In our case the stability group is SO(2) and we can write

eiωT3 |0〉 = eiω 1

4(a†a+aa†)|0〉eiω2k|0〉 = eiω/4|0〉. (104)

A new choice of parameters is r, s, ω, i.e.

g(α, β, γ) →

g(r, s, ω) =

[ √r 0

0 1/√r

] [

1 0s 1

][

cosω/2 − sinω/2sinω/2 cosω/2

]

(105)

where the new parameters are identified as

r = coshβ + sinhβ cosα

s = sinhβ sinα (106)

ω = α+ γ − arctan

(

sinα sinhβ/2

coshβ/2 + cosα sinhβ/2

)

(107)

An element of G can now be expressed as

g(r, s, ω) = eit1 ln reis(t3−t2)eiωt3 , (108)

and going to the unitary irreducible representation carried by even functionsthe coherent state becomes

|r, s〉 = T (r, s, 0)|0〉= eiT1 ln reis(T3−T2)e−

1

2 (xb )2

(109)

= exp

[

(is− 1)x2

r2b2

]

21

where Eq.(90) is used and b = 1/√µω. The parameters are related to average

values of the generators, such that

〈r, s|T3 − T2|r, s〉 =1

b√πr

∫ ∞

−∞

x2

2b2exp

(

−x2/rb2)

dx

=b3r

√πr

2b22b√πr

=r

4, (110)

and

〈r, s|T3|r, s〉 = 〈(

i

2x∂

∂x+i

4

)

〉

=1

b√πr

∫ ∞

−∞

[

ix2

2b2r(is− 1) +

i

4

]

exp(

−x2/rb2)

dx

= −s4, (111)

where s is real and r is real and positive (see M. Sugiara, Unitary Representations

and Harmonic Analysis, John Wiley&Sons, 1975).A convenient reparameterization of the wave packet in terms of u and w can

be accomplished as

b2r = 2w2, s = 2uw, (112)

so that the Gaussian wave packet becomes

ψ(x) ∝ exp

[

−(

1 − 2iuw

4w2

)

(x− q)2 + ipx

]

= |p, q, u, w〉. (113)

The Hamiltonian

H = − ~2

2µ

∂2

∂x2+ V (x) (114)

yields the wave packet average energy

E(p, q, u, w) =p2

2µ+u2

2µ+ U(q, w), (115)

with ~ = 1 and

U(q, w) =1

8µw2+ (2π)−1/2

∫ ∞

−∞

e−1

2y2

V (wy + q)dy (116)

We can now apply the TDVP equations to study the propagation of thiswave packet. The elements of the dynamical metric are

ηrs = i

[

∂2

∂r′∂s− ∂2

∂r∂s′

]

lnS, (117)

22

with

S = 〈p′, q′, u′, w′|p, q, u, w〉

=

∫ ∞

−∞

exp[

−ax2 + bx+ c]

dx

= eceb2/4a

∫ ∞

−∞

exp[

−(√ax− b/2

√a)2]

dx

= eceb2/4a√

π/a (118)

and where

a =

[

1 + 2iu′w′

4w′2+

1 − 2iuw

4w2

]

b = 2

[

1 + 2iu′w′

4w′2q′ +

1 − 2iuw

4w2q − ip′ + ip

]

c = −[

1 + 2iu′w′

4w′2q′2 +

1 − 2iuw

4w2q2]

. (119)

Differentiation of

lnS = c+ b2/4b− 1

2ln a+

1

2lnπ (120)

yields the elements of the upper triangle

ηpq = 1, ηpu = 0, ηpw = 0, ηqu = 0, ηqw = 0, ηuw = 1, (121)

in the antisymmetric metric matrix {ηrs}. The TDVP equations then become

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

pquw

=

∂E/∂p∂E/∂q∂E/∂u∂E/∂w

, (122)

or in more detail

q =p

µ,

p = −∂U∂q

,

w =u

µ,

u = −∂U∂w

, (123)

which look very much like the classical Hamilton’s equations.

23

4.2 The determinantal coherent state for N electrons

For anN−electron system we choose a set ofN spin orbitals u· = {u1, u2, · · · , uN}and form a determinantal wave function

det {u1(x1)u2(x2) · · ·uN(xN )} , (124)

or in second quantization the state vector

|0〉 =

N∏

i=1

a†i |vac〉, (125)

with |vac〉 the true vacuum state. We call this the reference state. The basis(of rank K) and the associated field operators are divided into two sets, thosethat refer to the reference state denoted by a• and the rest denoted by a◦, i.e.

(a•, a◦), (126)

and of course a similar partition of the creators, so that the reference state is

|0〉 =N∏

k=1

a•†k |vac〉. (127)

The creation operators and the basis transform in the same manner, so whenwe apply a general unitary transformation to the basis, the creation operatorssuffer the same transformation. We can write in matrix form

(b•†,b◦†) = (a•†, a◦†)

(

U• U′

U′′ U◦

)

, (128)

and conclude that the reference state becomes

N∏

i=1

b•†i |vac〉

=

N∏

i=1

N∑

l=1

a•†l U·li +

K∑

j=N+1

a◦†j U′′ji

|vac〉

=

N∏

i=1

N∑

l=1

{a•†l +

K∑

j=N+1

N∑

k=1

a◦†j U′′jk

(

U•−1)

kl}U•

li

|vac〉

= α

N∏

i=1

a•†i +

K∑

j=N+1

N∑

k=1

a◦†j U′′jk

(

U•−1)

ki

|vac〉

24

= α

N∏

i=1

1 +

K∑

j=N+1

N∑

k=1

a◦†j U′′jk

(

U•−1)

kia•i

a•†i | vac〉

= αN∏

i=1

1 +K∑

j=N+1

N∑

k=1

a◦†j U′′jk

(

U·−1)

kia•i

N∏

l=1

a•†l |vac〉. (129)

If we introduce the complex parameters

zji =N∑

k=1

U ′′jk

(

U•−1)

ki, (130)

and write the unnormalized state vector

N∏

i=1

b•†i |vac〉 = |z〉 (131)

with the parameters being time-dependent. In terms of the orthonormal spinorbital basis we write

|z〉 =

N∏

i=1

1 +

K∑

j=N+1

zjia◦†j a

•i

|0〉

=

N∏

i=1

K∏

j=N+1

[

1 + zjia◦†j a

•i

]

|0〉

=

N∏

i=1

K∏

j=N+1

exp[

zjia◦†j a

•i

]

|0〉

= exp

N∑

i=1

K∑

j=N+1

zjia◦†j a

•i

|0〉. (132)

In going from the second to the third line in the above equation we have usedthe fact that the electron field operators are nilpotent and this also is the reasonthat the exponentiation in the fourth line works. The end result is true becausethe all the operators a◦†j a

•i commute.

From these equations it follows straightforwardly that the wave functionrepresentative of this state is

det [χi(xj)] , (133)

with the ”dynamical spin orbitals”

χi = ui +

K∑

j=N+1

ujzji, (134)

25

and where the parameters zji are complex and are considered to be functions ofthe time- parameter t. As they change during a process involving the electronicsystem the determinantal state vector can in principle become any determinan-tal wave function possible to express in the spin orbital basis. The spin orbitalsχi are not orthonormal even if the basis {uk} is and in actual application onewill often use a raw basis of atomic spin orbitals (often built from Gaussian typeorbitals), which are not orthonormal. In such a case the exponential form of thedeterminantal state is not applicable and one has to deal with the full complica-tions of the non-unit metric of the basis (see Rev. Modern Phys. 66, 917 (1994)).This is indeed possible and has been coded into the ENDyne program systemthat uses narrow wave packet nuclei and single determinantal electron states inan explicitly time-dependent, nonadiabatic treatment of molecular processes.

A determinantal wave function expressed in this form is a coherent state.The associated Lie group is the unitary group U(K) and the reference stateEq.(127) is the lowest weight state of the irreducible representation [1N0K−N ]of U(K). The stability group is U(N)×U(K−N). The norm in an orthonormalbasis of spin orbitals is

〈z|z〉 = det[

1 + z†z]

, (135)

which means that we can define an appropriate measure dz in the parameterspace such that the resolution of the identity

∫

|z〉〈z|dz = 1 (136)

holds. The derivation of the form of the measure dz is nontrivial [12].

5 Minimal Electron Nuclear Dynamics (END)

The time-dependent variational principle introduced above with the quantummechanical action

A =

∫ t2

t1

Ldt, (137)

and the quantum mechanical Lagrangian (~ = 1)

L = 〈ψ| i2

(

∂

∂t− ∂

∂t

)

−H |ψ〉/〈ψ|ψ〉 (138)

proceeds by making the action stationary, which leads to the Euler-Lagrangeequations

d

dt

∂L

∂q=∂L

∂q, (139)

once the dynamical variables q have been chosen. The Electron Nuclear Dynam-ics (END) theory starts with this principle of least action, chooses a particular

26

family of wave functions ψ and a basis set for its description and derives a setof Euler-Lagrange equations as the dynamical equations that for that choice ofwave function form and basis represent the time-dependent Schrodinger equa-tion and can be integrated in time to yield the time-evolving state vector or thesystem under consideration.

A simple choice of wave function family for a molecular system that makessense is a product state vector

|ψ〉 = |z,R, P 〉|R,P 〉 = |z〉|φ〉 (140)

with |z,R, P 〉 = |z〉 an electronic state vector, and |φ〉 = |R,P 〉 a nuclear statevector. It should be noted that this product form is not an adiabatic type wavefunction since the electronic part is parametrically dependent on the (time-dependent) average nuclear positions R (and average momenta P ), not theiractual position (and momentum) coordinates. This corresponds rather to aBorn-Huang type product and is more like a diabatic wave function and couldin principle be resolved into a number of adiabatic states.

This lowest of END approximations aims at a description in terms of classicalnuclei, so a starting nuclear wave function is

|φ〉 =∏

j,k

exp[−1

2

(

Xjk −Rjk

b

)2

+ iPjk (Xjk −Rjk)], (141)

i.e. of product form, where the parameters Rjk and Pjk are the Cartesiancomponents of the average nuclear positions and momenta, respectively. Thesimplest electronic wave function that makes any sense is a complex spin unre-stricted Thouless determinant [13, 14]

|z,R, P 〉 ≡ |z〉 = det{χh(xp)}, (142)

with

χh = uh +∑

p

upzph, 1 ≤ h ≤ N, (143)

where {ui}K1 is a set of atomic spin orbitals (GTO’s) centered on the average

nuclear positions

~Rk =(

R1k R2k R3k

)

. (144)

In particular for high collision energies it is advantageous to include in thisatomic basis so called electronic translation factors exp{i~k·~r}, with ~k = m~Pk/Mk

for nucleus k to produce traveling atomic orbitals. The Lagrangian can now bewritten as

27

L =i

2[〈φ| ∂

∂t|φ〉/〈φ|φ〉 − 〈φ| ∂

∂t|φ〉/〈φ|φ〉

+ 〈z| ∂∂t

|z〉/〈z | z〉 − 〈z| ∂∂t

|z〉/〈z|z〉]− 〈ψH |ψ〉/〈ψ|ψ〉. (145)

We then use the fact that the parameters z,R, and P are considered to betime-dependent and employ the chain rule of differentiation to write

L =i

2

∑

i,l

{[〈φ| − ∂

∂Xil|φ〉/〈φ|φ〉

+ 〈φ| − ∂

∂Xil|φ〉/〈φ|φ〉 +

∂ lnS

∂Ril− ∂ lnS

∂R′il

]Ril

+ [∂ lnS

∂Pil− ∂ lnS

∂P ′il

]Pil} +i

2

∑

p,h

(

∂ lnS

∂zphzph − ∂ lnS

∂z∗ph

z∗ph

)

− 〈ψ|H |ψ〉/〈ψ|ψ〉, (146)

where we have used the form of the nuclear wave function, which is such thatthe differentiation with respect to Ril is the negative of the differentiation withrespect to the nuclear coordinate Xil. We have also introduced the overlapbetween two electronic wave functions S = 〈z|z〉 = S(z∗, R′, P ′; z,R, P ).

The narrow wave packet limit (b → 0, ~ → 0, ~/b2 → 1) leads to the La-grangian

L =∑

j,l

{(Pjl +i

2[∂ lnS

∂Rjl− ∂ lnS

∂R′jl

])Rjl +i

2[∂ lnS

∂Pjl− ∂ lnS

∂P ′jl

]Pjl}

+i

2

∑

p,h

(

∂ lnS

∂zphzph − ∂ lnS

∂z∗ph

z∗ph

)

−E, (147)

where

E =∑

j,l

P 2jl

2Ml− 〈z|Hel|z〉

〈z|z〉 , (148)

and contains the nuclear-nuclear repulsion terms, and where in the derivativesR′, P ′ are put equal to R, and P , respectively, after differentiation. When thetotal Hamiltonian is written

H = Tn + Te + Vne + Vee + Vnn (149)

with the nuclear kinetic energy terms Tn, the electron kinetic energy termsTe, the nuclear-electron interaction terms Vne, the electron-electron interaction

28

terms Vee, and the nuclear-nuclear interaction terms Vnn one can discern theelectronic Hamiltonian

Hel = Te + Vne + Vee + Vnn. (150)

Using this Lagrangian for quantum electrons and classical nuclei and choos-ing as the dynamical variable q = Rik we get

∂L

∂Rik

= Pik +i

2

[

∂ lnS

∂Rik− ∂ lnS

∂R′ik

]

, (151)

and

d

dt

∂L

∂Rik

= Pik +i

2

∑

j,l

[

∂2 lnS

∂Rik∂Rjl+

∂2 lnS

∂Rik∂R′jl

− ∂2 lnS

∂R′ik∂Rjl

− ∂2 lnS

∂R′ik∂R

′jl

]

Rjl

+i

2

∑

j,l

[

∂2 lnS

∂Rik∂Pjl+

∂2 lnS

∂Rik∂P ′jl

− ∂2 lnS

∂R′ik∂Pjl

− ∂2 lnS

∂R′ik∂P

′jl

]

Pjl

+i

2

∑

p,h

[

∂2 lnS

∂Rik∂zphzph − ∂2 lnS

∂R′ik∂zph

zph +∂2 lnS

∂Rik∂z∗ph

z∗ph − ∂2 lnS

∂R′ik∂z

∗ph

z∗ph

]

. (152)

This expression should equal

∂L

∂Rik=i

2

∑

p,h

[

∂2 lnS

∂Rik∂zphzph +

∂2 lnS

∂R′ik∂zph

zph − ∂2 lnS

∂Rik∂z∗ph

z∗p − ∂2 lnS

∂R′ik∂z

∗ph

z∗ph

]

+i

2

∑

j,l

[

∂2 lnS

∂Rik∂Rjl− ∂2 lnS

∂Rik∂R′jl

+∂2 lnS

∂R′ik∂Rjl

− ∂2 lnS

∂R′ik∂R

′jl

]

Rjl

+i

2

∑

j,l

[

∂2 lnS

∂Rik∂Pjl− ∂2 lnS

∂Rik∂P ′jl

+∂2 lnS

∂R′ik∂Pjl

− ∂2 lnS

∂R′ik∂P

′jl

]

Pjl

− ∂E

∂Rik, (153)

and cancellation of terms leads to the following result

Pik + 2∑

j,l

[

={

∂2 lnS

∂R′ik∂Rjl

}

Rjl + ={

∂2 lnS

∂R′ik∂Pjl

Pjl

}]

−i∑

p,h

[

∂2 lnS

∂Rik∂zphzph − ∂2 lnS

∂Rik∂z∗ph

z∗ph

]

= − ∂E

∂Rik(154)

29

or by collecting the three Cartesian components for each nucleus (such as Rk =(R1k, R2k, R3k) and the z parameters in a rectangular matrix z = {zph} we canwrite the more compact form 1

−Pk +∑

l

[

CRkRlRl + CRkPl

Pl

]

+ iC†Rk

z−iCTRk

z∗=~∇RkE, (155)

where

(CRR)ik;jl = (CRkRl)ij = −2= ∂2 lnS

∂R′ik∂Rjl

|R′=R,P ′=P , (156)

and

(CR)ph;ik = (CRk)ph;i = (CRik

)ph =∂2 lnS

∂z∗ph∂Rik|R′=R,P ′=P . (157)

Similarly we obtain for q = Pk we obtain

Rk +∑

l

[

CPkRlRl + CPkPl

Pl

]

+ iC†Pk

z − iCTPk

z∗ = ~∇PkE, (158)

and for q = z∗p the equation is

iCz + iCRR + iCPP =∂E

∂z∗. (159)

The complex conjugate of this equation obtains when we choose q = zp. Allthese equations of motion can be put together in matrix form

iC 0 iCR iCP

0 −iC∗ −iC∗R

−iC∗P

iC†R

−iCTR

CRR −I + CRP

iC†P

−iCTP

I + CPR CPP

z

z∗

R

P

=

∂E/∂z∗

∂E/∂z∂E/∂R∂E/∂P

. (160)

The integration of this set of coupled first-order differential equation can bedone in a number of ways. Care must be taken since there are basically tworather different time scales involved, i.e. that of the nuclear dynamics and thatof the normally considerably faster electron dynamics. It should be observedthat this electron nuclear dynamics takes place in a Cartesian laboratory refer-ence frame, which means that the overall translation as well as overall rotationof the molecular system is included. This offers no complications since the equa-tions of motion satisfy basic conservation laws and, thus, total momentum andangular momentum are conserved. At any time in the evolution of the molecu-lar system can the overall translation be isolated and eliminated if so should be

1One should note that (C†R

)ph;ik = (C∗R

)ik;ph = ∂2lnS/∂R′ik

∂zph and that (CTR

)ph;ik =(CR)ik;ph = ∂2 lnS/∂R′

ik∂z∗

ph.

30

deemed necessary. This level of theory [15, 14] is implemented in the programsystem ENDyne [16], and has been applied to atomic and molecular reactivecollisions. Calculations of cross sections, differential as well as integral, yieldresults in excellent agreement with the best experiments.

END theory at this level of approximation can be characterized as full non-linear time-dependent Hartree-Fock with moving classical nuclei. It is a direct,nonadiabatic approach to molecular processes. One might surmise that sucha method will do well at hyperthermal collision energies where many potentialsurfaces and associated nonadiabatic coupling terms will come into play. In par-ticular, ion-atom and ion-molecule reactions should be well described, since thedirect as well as the charge transfer processes can be treated on an equal footing.This is indeed true, and comparisons of calculated and measured direct absolute

cross sections for a number of systems at keV energies, such as H+, H, and Heon He and Ne (see [17, 18, 19]) have been published. Also, charge exchange,energy loss, and differential absolute cross sections for H+ and H on H andH2 in excellent agreement with the best experiments have been obtained withminimal END [20, 21]. The same is true for H+ on atomic nitrogen, oxygen,and fluorine [22], and N2 [23]

Differential cross sections and state-state processes are more sensitive tothe level of treatment than are integral cross sections, for which minimal ENDcan be shown to give results in good agreement with experiment also for lowerenergies, in some cases down to a fraction of an eV . Results for H+

2 on H2 [24],and H+ on C2H2 [25], and on C2H6 [26].

Cross sections and possible mechanisms have also been studied for the reac-tions [27, 28, 29]

D2 +NH+3 → NH3D

+ +D (161)

D2 +NH+3 → NH2D

+ +D +H (162)

with discovery of a two-step process for D → H exchange.That the simple minimal END wave function for the reacting system can

get such good results is puzzling to some and deserves a comment. Obviously,a single determinantal electronic wave function, even if it is complex and hasnonorthogonal spin orbitals, in a “static” time-independent calculation, can onlyyield results of SCF quality. However, in a time-dependent “dynamic” calcula-tion with the wave function adjusting at each time step to the moving nucleisuch a simple representation of the electrons provides a surprisingly flexibledescription of the electron dynamics.

The classical description of the nuclei means that a product molecular speciesvibrates and rotates as a classical object. Using the notion that coherent statesbridges the classical and quantum descriptions, one can view this classical mo-tion as an evolving state, which can be resolved into rovibrational states. In thecase of low excitation energies, when to a good approximation each vibrationalmode and the rotations can be considered to be decoupled, each normal vibra-tional mode can be represented by the evolving state in Eq.(75). The energy of

31

such a state is ~ω(|z|2 +1/2), yielding the vibrational excitation energy for sucha mode to be |z|2 = Evib/~ω, where the vibrational energy Evib is obtained froma so called generalized Prony analysis [30, 31, 32, 21]of the END trajectories. Inthis manner vibrationally resolved cross sections can be calculated a posteriori.A similar treatment can be given the rotational motion via another choice ofcoherent states [33].

The END theory is a general approach [34] to approximate the time-dependentSchrodinger equation and is not limited to treatments of molecular reaction dy-namics. Other problems that have been studied include intramolecular electrontransfer [14], the effect of intense laser light on the vibrational dynamics of smallmolecules [35], and solitonic charge transport in polyenes [36].

6 Rendering of Dynamics

When the forces between reactants are derived from a pre-calculated potentialenergy surface it is possible to produce informative pictures with reactant valleysand product valleys perhaps connected by saddles indicating transition states.Time-laps photography or movies of dynamical events may show probabilitiesin terms of nuclear wavefunctions evolving on one surface and then transfer toanother surface if nonadiabatic coupling terms are present.

In the case of direct nonadiabatic dynamics the rendering of dynamicalevents needs some rethinking. When the nuclei are treated as classical par-ticles or by narrow wave packets movies of dynamically changing ball and stickmodels can be quite effective and informative. It is possible to use the ren-dering of such trajectories for finding errors in the dynamics and to illustratemechanisms.

When the participating electronic degrees of freedom are treated dynami-cally, rather than being integrated out to provide the average forces, then onecan augment the rendering with an evolving electronic charge density of thereacting system. Sometimes a simple picture, such as is provided by a spher-ical electron cloud around each atom, the size of which changes in time withthe electron population, is useful and provides the crucial information aboutthe studied process. This manner of depicting the reacting system is shown inFig.1.

32

Figure 1: Six snap shots of H+ + C2H6 → CH4 + CH+3 with the dynamical

electrons represented by a sphere around each nucleus with the size proportionalto the electronic population on each atom. The H+ approaches from above inthe first frame, and polarizes the C − C bond in the second frame. The thirdthrough sixth frames show the products departing rovibrationally excited.

33

7 Acknowledgments

We would like to thank Remigio Cabrera-Trujillo for providing the figures, andwe acknowledge support from the Office of Naval Research.

References

[1] R. D. Levine, R. B. Bernstein, Molecular Reaction Dynamics and ChemicalReactivity, Oxford University, New York, 1987.

[2] W. H. Miller, Recent advances in quantum mechanical reactive scatteringtheory, Annu. Rev. Phys. Chem. 41 (1990) 245–281.

[3] J. M. Bowman, Overview of reduced dimensionality quantum approachesto reactive scattering, Theor. Chem. Acc. 108 (2002) 125–133.

[4] A. Froman, Isotope effects and electronic energy in molecules, J. Chem.Phys 36 (1962) 1490.

[5] M. Kimura, N. F. Lane, The low-energy, heavy-particle collisions - a close-coupling treatment, in: D. Bates, B. Bederson (Eds.), Advances in Atomic,Molecular and Optical Physics, Academic, New York, 1990, p. 79.

[6] J. Broeckhove, L. Lathouwers, E. Kesteloot, P. Van Leuven, On the equiv-alence of time dependent variational principles., Chem. Phys. Lett. 149(1988) 547.

[7] P. A. M. Dirac, The Principles of Quantum Mechanics, no. 27 in TheInternational Series of Monographs on Physics, Oxford University Press,1930.

[8] P. Kramer, M. Saraceno, Geometry of the Time-Dependent VariationalPrinciple in Quantum Mechanics, Springer, New York, 1981.

[9] Y. Ohrn, E. Deumens, Electron nuclear dynamics with coherent states,in: D. H. Feng, J. R. Klauder (Eds.), Proceedings of the InternationalSymposium on Coherent States: Past, Present, and Future, Oak RidgeAssociated Universities, World Scientific, Singapore, 1993.

[10] J. R. Klauder, B.-S. Skagerstam, Coherent States, Applications in Physicsand Mathematical Physics, World Scientific, Singapore, 1985.

[11] A. M. Perelomov, Coherent States for Arbitrary Lie Group, Commun.Math. Phys. 26 (1972) 222–236.

[12] J. P. Blaizot, H. Orland, Path integrals for the nuclear many-body problem,Phys. Rev. C 24 (1981) 1740.

[13] D. J. Thouless, Stability Conditions and Nuclear Rotations in the Hartree-Fock Theory., Nucl. Phys. 21 (1960) 225.

34

[14] E. Deumens, A. Diz, R. Longo, Y. Ohrn, Time-Dependent TheoreticalTreatments of The Dynamics of Electrons and Nuclei in Molecular Systems,Rev. Mod. Phys 66 (3) (1994) 917–983.

[15] Y. Ohrn, E. Deumens, A. Diz, R. Longo, J. Oreiro, H. Taylor, Time evo-lution of electrons and nuclei in molecular systems, in: J. Broeckhove,L. Lathouwers (Eds.), Time-Dependent Quantum Molecular Dynamics,Plenum, New York, 1992, pp. 279–292.

[16] E. Deumens, T. Helgaker, A. Diz, H. Taylor, J. Oreiro, B. Mo-gensen, J. A. Morales, M. C. Neto, R. Cabrera-Trujillo, D. Jacquemin,ENDyne version 5 Software for Electron Nuclear Dynamics, Quan-tum Theory Project, University of Florida, Gainesville FL 32611-8435,http://www.qtp.ufl.edu/endyne.html (2002).

[17] R. Cabrera-Trujillo, J. R. Sabin, Y. Ohrn, E. Deumens, Direct differentialcross section calculations for ion-atom and atom-atom collisions in the keVrange, Phys. Rev. A 61 (2000) 032719.

[18] R. Cabrera-Trujillo, J. R. Sabin, Y. Ohrn, E. Deumens, Charge exchangeand threshold effects in the energy loss of slow projectiles, Phys. Rev. Lett.84 (2000) 5300.

[19] R. Cabrera-Trujillo, E. Deumens, Y. Ohrn, J. R. Sabin, Impact parameterdependence of electronic and nuclear energy loss of swift ions: H+ → Heand H+ → H , Nucl. Instr. Meth. B168 (2000) 484.

[20] R. Cabrera-Trujillo, Y. Ohrn, E. Deumens, J. R. Sabin, Trajectory andmolecular binding effects in stopping cross section for hydrogen beams onH2, J. Chem. Phys. 116 (2002) 2783.

[21] J. A. Morales, A. C. Diz, E. Deumens, Y. Ohrn, Electron nuclear dynamicsof H++H2 collisions at Elab = 30eV , J. Chem. Phys 103 (1995) 9968–9980.

[22] R. Cabrera-Trujillo, Y. Ohrn, E. Deumens, J. R. Sabin, Stopping cross sec-tion in the low to intermediate energy range: Study of proton and hydrogenatom collisions with atomic N,O, and F , Phys. Rev. A 62 (2000) 052714.

[23] R. Cabrera-Trujillo, Y. Ohrn, E. Deumens, J. R. Sabin, B. G. Lindsay,Theoretical and experimental studies of the H+ +N2 sytem: Differentialcross sections for direct and charge-transfer scattering at keV energies,Phys. Rev. A 66 (2002) 042712.

[24] Y. Ohrn, J. Oreiro, E. Deumens, Bond making and bond breaking in molec-ular dynamics, Int. J. Quantum Chem. 58 (1996) 583.

[25] S. A. Malinovskaya, R. Cabrera-Trujillo, J. R. Sabin, E. Deumens, Y. Ohrn,Dynamics of collisions of protons with acetylene molecules at 30 ev, J.Chem. Phys. 117 (2002) 1103.

35

[26] R. Cabrera-Trujillo, J. R. Sabin, Y. Ohrn, E. Deumens, Energy loss studiesof protons colliding with ethane: preliminary results, J. Elec. Spectr. 129(2003) 303–308.

[27] M. Coutinho-Neto, E. Deumens, Y. Ohrn, Abstraction and exchange mech-anisms for the D2 + NH+

3 rea ction at hyperthermal energies, J. Chem.Phys. 116 (2002) 2794.

[28] R. J. S. Morrison, W. E. Conway, T. Ebata, R. N. Zare, Vibrationallystate-selected reactions of ammonia ions. i. NH+

3 (v)+D2, J. Chem. Phys.84 (1986) 5527.

[29] J. C. Poutsma, M. A. Everest, J. E. Flad, G. C.Jone s Jr., R. N. Zare, State-selected studies of the reaction of NH+

3 (v1, v2) with D2, Chem. Phys. Lett.305 (1999) 343.

[30] B. de Prony, U, J. E. Polytech. 1 (2) (1795) 24–76.

[31] A. Blass, E. Deumens, Y. Ohrn, Rovibrational analysis of molecular colli-sions using coherent states, J. Chem. Phys. 115 (2001) 8366.

[32] J. A. Morales, A. C. Diz, E. Deumens, Y. Ohrn, Molecular VibrationalState Distributions in Collisions, Chem. Phys. Lett. 233 (1995) 392.

[33] J. A. Morales, E. Deumens, Y.Ohrn, On rotational coherent states in molec-ular quantum dynamics, J. Math. Phys. 40 (1999) 766.

[34] Y. Ohrn, E. Deumens, Towards an ab initio treatment of the time-dependent scroding er equation of molecular systems, J. Phys. Chem. 103(1999) 9545.

[35] J. Broeckhove, M. D. Coutinho-Neto, E. Deumens, Y. Ohrn, Electron nu-clear dynamics of LiH and HF in an intense laser field, Phys. Rev. A 56(1997) 4996.

[36] B. Champagne, E. Deumens, Y. Ohrn, Vibrations and soliton dynamics ofpositively charged polyacetylene chains, J. Chem. Phys. 107 (1997) 5433.

36