Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

192

satrajit adhikari and gert due billing

coordinates

RAB2 ¼ ðrx2 þ ry2Þd12

R2

¼ ð

R2

R2

Þ

d22

ð1 cos x2Þ

r2

þ

r2

Þ

d22ð1 þ cos x2Þ

BC

x þ

y

2

þ ð x

y

2

ðrxRx þ ryRyÞd22 sin x2

ðC:3Þ

R2

¼ ð

R2

R2

Þ

d32

ð1 cos x3Þ

r2

þ

r2

Þ

d32ð1 þ cos x3Þ

CA

x þ

y

2

þ ð x

y

2

þ ðrxRx þ ryRyÞd32 sin x3

where dk2 ¼ ðmk=mÞð1 mk=MÞ, m1 m2 and m3 are the masses of the atom A, B,

and C, respectively, in the corners of the triangle ABC. The parameters p

M ¼ m1 þ m2 þ m3 and m ¼ m1m2m3=M and the angles are given by x2 ¼

2arctan ðm3=m) and x3 ¼ 2 arctan ðm2=m). By using Eq. (C.2) one can write

R2

R2

cos x2

R2

R2

cos x3

R2

R2

CA

AB

BC

AB

BC

CA

d2

d2

d2

d2

d2

d2

3

2

1

2

3

tan f

¼

!

1R"2

!

R2"

þ !

"

C:4

Þ

R2

R2

!

AB

CA

" sin

x2 !

BC

AB

" sin

x3

ð

d12

d32

d22

d12

It would be convenient for obtaining the expressions of the gradient of the hyperangle f with respect to Jacobi coordinates to introduce the physical region of the conical intersection in the following manner:

qf

¼

qf

qRAB

þ

qf

qRBC

þ

qf

qRCA

qri

qRAB

qri

qRBC

qri

qRCA

qri

ðC:5Þ

qf

¼

qf

qRAB

þ

qf

qRBC

þ

qf

qRCA

qRi

qRAB

qRi

qRBC

qRi

qRCA

qRi

where i x; y; z. To obtain explicit expressions for 5f, we have used Eqs. (C.2–C.5) and after some algebra (!) it is interesting to note that 5f becomes independent of dk and xk for any arbritrary A þ BC type reactive system. We obtain

	qf	2					ðri sin f þ Ri cos fÞ
		¼
	qri				r2 sin y
qf		2				ð ri cos f þ Ri sin fÞ
			¼
qRi				r2 sin y
	qf	¼ 0					ðC:6Þ
qrz
qf			¼ 0
qRz

non-adiabatic effects in chemical reactions

193

where i x; y. Similarly, explicit expressions for 5y are obtained using Eqs. (C.1)

	qy			2Ry
		¼
qrx					r2
	qy			2Rx
		¼
qry					r2
	qy		¼	2ry
qRx				r2			ðC:7Þ
	qy		¼	2rx
qRy				r2
	qy		¼ 0
	qrz
	qy		¼ 0
qRz

The azimuthal angle (Z) about the conical intersection is related with hyperangles y and f as

		sin y sin f
Zðy; fÞ f0	¼ arctan			ðC:8Þ
		cos y0 sin y cos f þ sin y0 cos y

where y0 indicates the position of the conical intersection.

The gradient of 5Z with respect to Jacobi coordinates (the vector potential) considering the physical region of the conical intersection, is obtained by using Eqs. (C.6–C.8) and after some simplification (!) we get,

	qZ	¼				2 ½Ry sin y0 sin f þ ðcos y0 sin y þ sin y0 cos y cos fÞðrx sin f þ Rx cos fÞ&
	qrx			r2				½sin2 y sin2 f þ ðcos y0 sin y cos f þ sin y0 cos yÞ2&
	qZ	¼				2 ½Rx sin y0 sin f þ ðcos y0 sin y þ sin y0 cos y cos fÞðry sin f þ Ry cos fÞ&
	qry			r2				½sin2 y sin2 f þ ðcos y0 sin y cos f þ sin y0 cos yÞ2&
	qZ	¼ 0
	qrz
qZ		¼	2 ½ry sin y0 sin f þ ðcos y0 sin y þ sin y0 cos y cos fÞð rx cos f þ Rx sin fÞ&
qRx			r2					½sin2 y sin2 f þ ðcos y0 sin y cos f þ sin y0 cos yÞ2&
qZ		¼	2 ½rx sin y0 sin f þ ðcos y0 sin y þ sin y0 cos y cos fÞð ry cos f þ Ry sin fÞ&
qRy			r2					½sin2 y sin2 f þ ðcos y0 sin y cos f þ sin y0 cos yÞ2&
qZ		¼ 0						ðC:9Þ
qRz

194 satrajit adhikari and gert due billing

For the H3 isotopic variants, we can calculate the values of y0 and f0 by introducing RAB ¼ RBC ¼ RCA. Moreover, we get y0 ¼ f0 ¼ 0 for an A3 and f0 ¼ 0 for an AB2 type reactive system. In case of an A þ B2 type reaction, one can use

sin y0		d12					C:10
			d22
							ð Þ
					d12
	¼ d22 cos x2

and obtain y0 ¼ 11:5 for DH2 and y0 ¼ 14:5 for HD2. The actual position of the CI on the PES is obtained through the equation, Vðr0; y0; f0Þ ¼ ECI where ECI is the potential energy at the point of the CI.

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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)

COMPLEX STATES OF SIMPLE

MOLECULAR SYSTEMS

R. ENGLMAN

Department of Physics and Applied Mathematics, Soreq NRC,

Yavne, Israel;

College of Judea and Samaria, Ariel, Israel

A. YAHALOM

College of Judea and Samaria, Ariel, Israel

CONTENTS

I.Introduction and Preview

II. Aspects of Phase in Molecules

III. Analytic Theory of Complex Component Amplitudes

A.Modulus and Phase

B.Origin of Reciprocal Relations

1.A General Wavepacket

C.Other Phase-Modulus Relations

1.The Equation of Continuity

2.The WKB Formula

3.Extended Systems

4.Loss of Phase in a Quantum Measurement

D.The Cauchy-Integral Method for the Amplitudes

E.Simplified Cases

1.The Near-Adiabatic Limit

2.Cyclic Wave Functions

3.Wave Packets

F.Consequences

197

198	r. englman and a. yahalom

IV. Nonlinearities That Lead to Multiple Degeneracies

A.Conical Intersection Pairs

1.1A1 and 2A2 States in C2v Symmetry

B.Continuous Tracing of the Component Phase

1.A1 and B2 States in C2v Symmetry

2.Trigonal Degeneracies

C.The Adiabatic-to-Diabatic Transformation (ADT)

D.Direct Integration

E.Higher Order Coupling in Some Jahn–Teller and Renner–Teller Effects

1.Complex Representation

2.Squaring of Off-Diagonal Elements

3.General Off-Diagonal Coupling

4.Nonlinear Diagonal Elements

5.Generalized Renner–Teller Coupling

6.Interpretation

F.Experimental Phase Probing V. Molecular Yang–Mills Fields

A.A Nuclear Lagrangean

B.Pure versus Tensorial Gauge Fields

C.The ‘‘Curl Condition’’

D.The Untruncated Hilbert Space

E.An Alternative Derivation

F.General Implications

G.An Extended (Sufficiency) Criterion for the Vanishing of the Tensorial Field

H.Observability of Molecular States in a Hamiltonian Formalism

I.An Interpretation

VI. Lagrangeans in Phase-Modulus Formalism

A.Background to the Nonrelativistic and Relativistic Cases

B.Nonrelativistic Electron

C.Similarities Between Potential Fluid Dynamics and Quantum Mechanics

D.Electrons in the Dirac Theory

E.The Nearly Nonrelativistic Limit

F.The Lagrangean-Density Correction Term

G.Topological Phase for Dirac Electrons

H.What Have We Learned About Spinor Phases?

VII. Conclusion

Acknowledgments

References

I.INTRODUCTION AND PREVIEW

In quantum theory, physical systems move in vector spaces that are, unlike those in classical physics, essentially complex. This difference has had considerable impact on the status, interpretation, and mathematics of the theory. These aspects will be discussed in this chapter within the general context of simple molecular systems, while concentrating at the same time on instances in which the electronic states of the molecule are exactly or nearly degenerate. It is hoped

complex states of simple molecular systems

199

that as the chapter progresses, the reader will obtain a clearer view of the relevance of the complex description of the state to the presence of a degeneracy.

The difficulties that arose from the complex nature of the wave function during the development of quantum theory are recorded by historians of science [1–3]. For some time during the early stages of the new quantum theory the existence of a complex state defied acceptance ([1], p. 266). Thus, both de Broglie and Schro¨dinger believed that material waves (or ‘‘matter’’ or ‘‘de Broglie’’ waves, as they were also called) are real (i.e., not complex) quantities, just as electromagnetic waves are [3]. The decisive step for the acceptance of the complex wave came with the probabilistic interpretation of the theory, also known as Born’s probability postulate, which placed the modulus of the wave function in the position of a (and, possibly, unique) connection between theory and experience. This development took place in the year 1926 and it is remarkable that already in the same year Dirac embraced the modulus-based interpretation wholeheartedly [4]. Oddly, it was Schro¨dinger who appears to have, in 1927, demurred at accepting the probabilistic interpretation ([2], p. 561, footnote 350). Thus, the complex wave function was at last legitimated, but the modulus was and has remained for a considerable time the focal point of the formalism.

A somewhat different viewpoint motivates this chapter, which stresses the added meaning that the complex nature of the wave function lends to our understanding. Though it is only recently that this aspect has come to the forefront, the essential point was affirmed already in 1972 by Wigner [5] in his famous essay on the role of mathematics in physics. We quote from this here at some length:

‘‘The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for. . . this uncanny usefulness of mathematical concepts. . .

The complex numbers provide a particularly striking example of the foregoing. Certainly, nothing in our experience suggests the introducing of these quantities. . . Let us not forget that the Hilbert space of quantum mechanics is the complex Hilbert space with a Hermitian scalar product. Surely to the unpreoccupied mind, complex numbers. . . cannot be suggested by physical observations. Furthermore, the use of complex numbers is not a calculational trick of applied mathematics, but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now (1972) begins to appear that not only complex numbers but analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations. It is difficult to avoid the impression

that a miracle confronts us here [i.e., in the agreement between the properties of p

the hypernumber ð1Þ and those of the natural world].’’

A shorter and more recent formulation is ‘‘The concept of analyticity turns out to be astonishingly applicable’’ ([6], p. 37).

200	r. englman and a. yahalom

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of their own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as ‘‘miraculous.’’

A term that is nearly synonymous with complex numbers or functions is their ‘‘phase.’’ The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac’s terminology [7], which writes a wave function by a set of coefficients, the ‘‘amplitudes,’’ each expressible in terms of its absolute value, its ‘‘modulus,’’ and its ‘‘phase.’’) There is a related growth of literature on interference effects, associated with Aharonov–Bohm and Berry phases [8–14]. In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techniques to achieve these are also anchored in the phases of the wave function components. This trend is manifest in such diverse areas as coherent or squeezed states [15,16], electron transport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23]. Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar methods [25].

The status of the phase in quantum mechanics has been the subject of debate. Insomuch as classical mechanics has successfully formulated and solved problems using action-angle variables [26], one would have expected to see in the phase of the wave function a fully ‘‘observable’’ quantity, equivalent to and having a status similar to the modulus, or to the equivalent concept of the ‘‘number variable’’. This is not the case and, in fact, no exact, well-behaved Hermitean phase operator conjugate to the number is known to exist. (An article by Nieto [27] describes the early history of the phase operator question, and gives a feeling of the problematics of the field. An alternative discussion, primarily related to phases in the electromagnetic field, is available in [28]). In Section II, a brief review is provided of the various ways that phase is linked to molecular properties.

Section III presents results that the analytic properties of the wave function as a function of time t imply and summarizes previous publications of the authors and of their collaborators [29–38]. While the earlier quote from Wigner has prepared us to expect some general insight from the analytic behavior of the wave function, the equations in this section yield the specific result that, due to the analytic properties of the logarithm of wave function amplitudes, certain forms of phase changes lead immediately to the logical necessity of enlarging

complex states of simple molecular systems

201

the electronic set or, in other words, to the presence of an (otherwise) unsuspected state.

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schro¨dinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers–Kronig type of reciprocal relations. We propose that behind these relations there may be an ‘‘equation of restriction,’’ but while in the former case (where the variable is the frequency) the equation of restriction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40].

As already mentioned, the results in Section III are based on dispersions relations in the complex time domain. A complex time is not a new concept. It features in wave optics [28] for ‘‘complex analytic signals’’ (which is an electromagnetic field with only positive frequencies) and in nondemolition measurements performed on photons [41]. For transitions between adiabatic states (which is also discussed in this chapter), it was previously introduced in several works [42–45].

Interestingly, the need for a multiple electronic set, which we connect with the reciprocal relations, was also a keynote of a recent review ([46] and previous publications cited there and in [47]). Though the considerations relevant to this effect are not linked to the complex nature of the states (but rather to the stability of the adiabatic states in the real domain), we have included in Section III a mention of, and some elaboration on, this topic.

In further detail, Section III stakes out the following claims: For timedependent wave functions, rigorous conjugate relations are derived between analytic decompositions (in the complex t plane) of phases and of log moduli. This entails a reciprocity, taking the form of Kramers–Kronig integral relations (but in the time domain), holding between observable phases and moduli in several physically important cases. These cases include the nearly adiabatic (slowly varying) case, a class of cyclic wave functions, wavepackets, and noncyclic states in an ‘‘expanding potential.’’ The results define a unique phase through its analyticity properties and exhibit the interdependence of geometric phases and related decay probabilities. It turns out that the reciprocity property obtained in this section holds for several textbook quantum mechanical applications (like the minimum width wavepacket).