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The Role of Degenerate States in Chemistry: Advances in Chemical Physics, Volume 124.

Edited by Michael Baer and Gert Due Billing. Series Editors I. Prigogine and Stuart A. Rice. Copyright # 2002 John Wiley & Sons, Inc.

ISBNs: 0-471-43817-0 (Hardback); 0-471-43346-2 (Electronic)

ELECTRON NUCLEAR DYNAMICS

¨

YNGVE OHRN and ERIK DEUMENS

University of Florida, Quantum Theory Project, Departments of

Chemistry and Physics, Gainesville, FL

CONTENTS

I.Introduction

II. Structure and Dynamics III. TDVP and END

A.The Basic END Ansatz

B.Free Electrons

C.General Electron Nuclear Dynamics IV. Molecular Processes

A.Reactive Collisions

B.Final-State analysis

C.Intramolecular Electron Transfer References

I.INTRODUCTION

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schro¨dinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes.

Traditionally, for molecular systems, one proceeds by considering the electronic Hamiltonian Hel, which consists of the quantum mechanical operators for the kinetic energy of the electrons, their mutual Coulombic repulsions, and

323

324	yngve o¨hrn and erik deumens

their attractions to each of the atomic nuclei. Commonly, the nuclear–nuclear repulsion energy is included in Hel. The time-dependent Schro¨dinger equation with this Hamiltonian describes the electron dynamics in a field of stationary nuclei. Methods of solving the time-independent electronic Schro¨dinger equation, commonly referred to as electronic structure theory, have reached considerable refinement and accuracy over the past decades. The bulk of such work consists of development of approximate many-electron theory and its implementation in terms of sophisticated computer software for solution of

Heljni ¼ EnðRÞjni

ð1Þ

that is, finding approximate stationary state solutions jni with characteristic electronic energies EnðRÞ for one fixed nuclear geometry at a time.

While experiments by their very nature are carried out in a laboratory coordinate frame, theory commonly proceeds via the introduction of internal coordinates in terms of molecule fixed axes. Done properly, this means that the kinetic energy of the center-of-mass motion is first separated from the other degrees of freedom. The origin of the internal coordinates, of course, can be chosen in a number of ways. The center of mass of the nuclei is a convenient choice that does not introduce kinetic energy coupling terms between electronic and nuclear degrees of freedom. No matter what is the choice of origin of the internal system of coordinates the result is a set of modified kinetic energy operators with reduced particle masses and so-called mass polarization terms. The latter, which are sums of products of momenta of different particles, are as a rule small and usually neglected.

For some systems consisting of two-to-four atoms of light elements it is currently feasible to consider enough points for, say, the ground-state electronic energy E0ðRÞ such that appropriate interpolation techniques can produce the energy for all nuclear geometries below some suitable energy cutoff. The resulting function E0ðRÞ is the Born–Oppenheimer (BO) potential energy surface (PES) of the system. Traditional molecular reaction dynamics proceeds by considering such a PES to be the potential energy for the nuclear dynamics, which, of course, may be treated classically, semiclassically, or by employing quantum mechanical methods. The other energy eigenvalues EnðRÞ similarly yield potential energy surfaces for electronically excited states. Each PES usually exhibits considerable structure for a polyatomic system and can provide instructive pictures with reactant and product valleys, local minima identifying stable species, and transition states providing gateways for the system to travel from one local minimum to another. Avoided crossings or more generally conical intersections and potential surface crossings are regions of dramatic chemical change in the system. The PES in this way provides attractive pictures of dynamical processes, which since the very beginning of molecular reaction dynamics have dominated our ways of thinking about molecular processes.

electron nuclear dynamics

325

Quantum mechanical methods using high-quality potential energy surfaces have produced results in excellent agreement with the best experiments for small systems of the lightest elements at low energies. However, high quality potential energy surfaces exist only for a few systems. The difficulty of their determination increases rapidly with the number of atoms in the system. The determination of a PES in 3n 6 dimensions for an n-atom system is not only costly, but a PES in six or more dimensions is very hard to visualize and thus less useful. One way to proceed for larger systems is to identify active modes and to freeze or discretize other degrees of freedom. Such procedures tend to be subjective, and may introduce artificial features into the dynamics. In this traditional approach to dynamics, each process is studied at a fixed total angular momentum.

Many molecular beam experiments are performed at collision energies from a fraction of an electron volt to tens of electron volts. In such cases two or more stationary molecular electronic states and their potential energy surfaces can provide an adequate description provided also the effects of the nonadiabatic coupling terms are taken into account. Even in cases where a single PES is sufficient to describe the relevant forces on the participating nuclei one should augment the Born–Oppenheimer PES with the diagonal kinetic energy correction to produce the so-called adiabatic approximation, something that is only rarely done in practice.

Ion–atom and ion–molecule collisions at energies in the kiloelectron volts range are common in studies of energy deposition and stopping of swift particles in various materials. Theoretical treatments of such processes often employ stationary electronic states and their potential energy surfaces. At such elevated energies the relevant state vector of the system is an evolving state, which may be expressed as a superposition of a number of such energy states. In fact, the system moves on an effective PES, which is the dynamical average of a number of adiabatic surfaces and should in principle also include effects of the nonadiabatic coupling terms.

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be just as useful. The electron nuclear dynamics (END) theory [1– 4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schro¨dinger equation are derived by employing the timedependent variational principle (TDVP).

II.STRUCTURE AND DYNAMICS

The most accurate information about quantum systems is obtained via spectroscopic measurements. Such measurements have, until quite recently,

326	yngve o¨hrn and erik deumens

been capable of only rather long-time averages of molecular events. Such studies emphasize structure and the associated electronic structure theory can successfully calculate molecular spectra and properties by applying the timeindependent Schro¨dinger equation and focusing on stationary electronic states. The methods and techniques of electronic structure theory have a long history, and coupled with the development of ever more powerful computers this area of study has reached a very high degree of sophistication.

Development of laser technology over the last decade or so has permitted spectroscopy to probe short-time events. Instead of having to resort to the study of reactants and products and their energetics and structures, one is now able to follow reactants as they travel toward products. Fast pulsed lasers provide snapshots of entire molecular processes [5] demanding similar capabilities of the theory. Thus, explicitly time-dependent methods become suitable theoretical tools.

The dominant theoretical approaches to study molecular processes break the problem down into separate parts, the first being the determination of one or more potential energy surfaces. This involves electronic structure calculations for a large number of nuclear geometries and interpolation techniques [6–8] to provide as accurate as possible a functional form of each PES. Electronic structure methods and algorithms have been developed into efficient codes such as Gaussian [9], and ACES II [10], which can be used with minimal knowledge of electronic structure theory. These and many other codes have made computational chemistry a working tool for the bench chemist on an equal footing with various spectroscopic methods.

Given a ground-state PES the dynamics methods that have dominated the field since the beginning proceed by treating the nuclear motion with quantum mechanics, semiclassical or quasiclassical techniques, or with classical trajectory methods. For processes where more than one electronic stationary state is involved one needs preconstructed PESs for all states and also nonadiabatic coupling terms in order to study the dynamics. Several workers have contributed significantly to the developments of a variety of methods for molecular dynamics with active electronic degrees of freedom [11–17]. The electronic basis employed for the evolution of approximate solutions to the time-dependent Schro¨dinger equation may consist of the electronic energy eigenstates and the nonadiabatic effects can be accounted for by calculated and interpolated coupling terms or simulated by phenomenological surface hopping.

A major drawback to the approaches that use preconstructed PESs is that there are many more interesting systems undergoing reactive dynamical processes than there are available PESs. It would be much better if the electronic structure part of the problem, which provides the forces for the nuclear dynamics, could be performed simultaneously with the dynamics part, thus making possible the treatment of systems for which preconstructed PESs

electron nuclear dynamics

327

do not exist. Such approaches are referred to as direct dynamics methods. The popularity of the Car–Parinello method [18] is ample evidence for the need of such theoretical dynamics treatments. Car–Parinello uses density functional theory for the electronic degrees of freedom and may be considered to be a direct dynamics method for processes that strictly follow the electronic ground state.

Potential energy surfaces, although purely theoretical constructs, are extremely useful and attractive tools in providing illustrative pictures of molecular processes and are essential for understanding the energetics. When the dynamics takes place on a single surface one can picture the dynamics moving from a reactant valley to a product valley passing through a transition state region. When several PESs are involved one tends to picture the dynamics as following one surface or another with the nonadiabatic coupling terms providing the means for transitions from one surface to the other. Strictly speaking, nonadiabatic dynamics takes place between PESs and one can very well use a different basis from that of electron energy eigenstates in describing the evolving system. The challenge is then to develop simple and illustrative alternative pictures of the molecular process to that provided by PESs.

Electron nuclear dynamics theory is a direct nonadiabatic dynamics approach to molecular processes and uses an electronic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the interparticle interactions.

III. TDVP AND END
The TDVP employs the quantum mechanical action
	t2
A ¼ ðt1			Lðc ; cÞdt	ð2Þ
where the Lagrangian is (h ¼ 1)
L ¼ hcji	q		Hjci=hcjci	ð3Þ
	qt

328	yngve o¨hrn and erik deumens

with H the total Hamiltonian of the system and requires the action to be stationary under variations of the wave function, that is,

t2
dA ¼ ðt1	dLðc ; cÞdt ¼ 0	ð4Þ

The total molecular system wave function is subject to the boundary conditions

djci ¼ dhcj ¼ 0 ð5Þ

at the endpoints t1 and t2.

When the wave function is completely general and permitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schro¨dinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schro¨dinger equation.

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational

wave function, and average nuclear positions and momenta. We write
jci jcðzÞi jzi	ð6Þ

where z denotes an array of suitable, and in general complex, wave function parameters.

By using Eq. (5), we can write the Lagrangian in a more symmetric form as

L ¼	2 ðhzjzi hzjziÞ hzjHjzi hzjzi		ð7Þ
		i

where the dot denotes differentiation with respect to the time parameter t. Variation of the Lagrangian, dL, with respect to all the parameters introduces jdzi and hdzj, which can be eliminated by the introduction of the total time derivatives

d hzjdzi

dt hzjzi

ð8Þ

electron nuclear dynamics

329

and

d

hdzjzi

ð

9

Þ

dt

h

z z

j i

and the boundary conditions Eq. (5). This results in a set of equations

t2

0 ¼ dA ¼ ðt1

dL dt

t2

X X

q2 ln S

qE

¼ ðt1

a

" b

i

qzaqzb

zb

qza

#dza

X X

q2 ln S

qE

þ

" b

i

zb

#dza!dt

ð10Þ

a

qzaqzb

qza

where S ¼ Sðz ; zÞ ¼ hzjzi and E ¼ Eðz ; zÞ ¼ hzjHjzi=hzjzi, and where the chain rule has been applied to the time differentiation.

Since dza and dza are independent variations it must follow that

	X		qE
i	Cabzb ¼		qz	ð11Þ
	b		a
	X		qE
i	C z				12
	ab b	¼ qza		ð		Þ
	b

where Cab ¼ q2 ln S=qzaqzb. These equations govern the time evolution of the wave function parameters. The time evolution of the overall phase factor exp ig is controlled by the equation

g ¼	i	X	za	q	za	q	ln S E	ð13Þ
	2 a			qza		qza

Note that for a stationary state all parameters satisfy z ¼ 0 and thus g ¼ Et, yielding the phase-factor exp iEt as expected.

In matrix block form the equations that govern the time evolution of the parameters can be expressed as

0 iC		z	¼	qqE=qqz	ð14Þ
iC	0	z		E= z

330	yngve o¨hrn and erik deumens

If the wave function parameters are chosen appropriately, then the Hermitian matrix C ¼ ½Cab& has an inverse and we can write

z	¼	i0	1	iC 1	qqE=qqz	ð15Þ
z		C		0	E= z

It is possible to introduce a generalized Poisson bracket by considering two general differentiable functions f ðz; z Þ and gðz; z Þ and write

ff ; gg ¼ ½qf =qzT qf =qzy&

0

1

iC 1

! qqg=qqz

iC

0

g=

z

¼ i

X

ðC 1Þab qqzb qqza ðC 1

Þab qqzb#

ð16Þ

a;b "qqzfa

g

g

f

It follows straightforwardly that

z ¼ fz; Eg

ð17Þ

z ¼ fz ; Eg

ð18Þ

which shows that the time evolution of the wave function parameters is governed by Hamilton-like equations. The time evolution of the molecular system can then be viewed as occurring on a phase space made up of the complex wave function parameters z and z acting as conjugate positions and momenta and Eðz; z Þ being the Hamiltonian or the generator of infinitesimal time translations. This is obviously not a flat phase space. Such coupled sets of first-order differential equations can be integrated by a great variety of methods (see [22]).

A.The Basic END Ansatz

The END theory can be implemented at various levels of approximation. The simplest approximation develops a Lagrangian for classical nuclei or distinguishable atomic nuclei represented by traveling Gaussian wavepackets in the narrow width limit, and for quantum electrons represented by a single determinant built from nonorthogonal, complex spin orbitals [23]. The principle of least action using this Lagrangian yields the dynamical equations of minimal END.

At this level of approximation, the molecular wave function can be expressed

as

j ðtÞi ¼ jRðtÞ; PðtÞijzðtÞ; RðtÞ; PðtÞi

ð19Þ

electron nuclear dynamics

where

k

2

Y

1

Xk Rk

h

X R t

; P t

Þi ¼

exp

" 2

b

þ

iP

k ð

X

R

j ð Þ

ð

k

kÞ#

and

hxjzðtÞ; RðtÞ; PðtÞi ¼ det wiðxjÞ

with the spin orbitals

X

wi ¼ ui þ

K

ujzjiðtÞ

j ¼ Nþ1

expanded in terms of atomic spin orbitals

fuigK1

which in turn are expanded in a basis of traveling Gaussians,

		i
ðx RxÞlðy RyÞmðz RzÞn exp aðx RÞ2			P ðx RÞ
		hM

331

ð20Þ

ð21Þ

ð22Þ

ð23Þ

ð24Þ

centered on the average nuclear positions R and moving with velocity P=M. In the narrow wavepacket limit, b ! 1, the Lagrangian may be expressed as

L ¼ i; j

("Pjl þ 2

qqRjl

qqRjl0

!#R_ jl þ 2

qqPjl

qqPjl0

!P_ jl)

X

i

ln S

ln S

i

ln S

ln S

i

X

ln S

q ln S

þ

p;h

q

zph

zph

E

ð25Þ

2

qzph

qz ph

with S ¼ hz; R0; P0jz; R; Pi and

X

z; R0; P0

z; R0; P0 z; R; P

E

¼

P2

=2M

l þ h

H z; R; P =

i

ð

26

Þ

jl

j elj

i h

j

jl

Here, Hel is the electronic Hamiltonian including the nuclear–nuclear repulsion terms, Pjl is a Cartesian component of the momentum, and Ml the mass of nucleus l. One should note that the bra depends on z while the ket depends on z and that the primed R and P equal their unprimed counterparts and the prime simply denotes that they belong to the bra.