Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

Atomic Clocks and Constraints on Variations of Fundamental Constants Laboratory Constraints on Time Variations 461

If one compares transitions of the same type – gross

structure, ﬁne structure – the ﬁrst term cancels.

30.3.1 Constraints from Absolute Optical

Measurements

Absolute frequency measurements offer the possibility

to compare a number of optical transitions with fre-

quencies f

, which scale as cR

∞

, with the caesium

hyperﬁne structure. One can rewrite (30.7)as

∂ln f

opt

∂t

∂ ln cR

∞

∂t

+ κ

∂lnα

∂t

(30.8)

where dimensional quantities, such as frequency and

the Rydberg constant, are stated in SI units (30.1). This

equation may be used in different ways. For example,

in Fig. 30.4 we plot experimental data for ∂ ln f

opt

/∂t

as a function of the sensitivity κ and derive a model-

independent constraint on the variation of the ﬁne

structure constant

∂lnα

∂t



− 0.3 ± 2.0×10

−15



−1

(30.9)

and the numerical value of the Rydberg frequency cR

∞

(Table 30.4) in the SI unit of Hertz. The latter is of

great metrological importance, being related to a com-

mon drift of optical clocks with respect to a caesium

clock, i. e., to the deﬁnition of the SI second. The

SI deﬁnition of the metre is unpractical and so, in prac-

tice, the optical wavelengths of reference lines calibrated

against the caesium standard are used to determine the

SI metre [30.37].

The constraints on the variations of α and cR

∞

are

correlated and the standard uncertainty ellipse, deﬁned





∂lnf

∂t

−

∂lnRy

∂t

− κ

∂lnα

∂t



= 1 + χ

min

is presented in Fig. 30.5. Here we sum over all available

data: ∂

(

ln f

)

/∂t is the central value of the observed drift

rate, u

its 1σ uncertainty, and χ

min

the minimized χ

of the ﬁt.

Table 30.4 Model-independent laboratory constraints on

the possible time variations of natural constants

X ∂lnX/∂t

α ( − 0.3 ± 2.0) ×10

−15

−1

{cR

∞

} ( − 2.1 ± 3.1) ×10

−15

−1

/µ

(3.0 ± 6.8) ×10

−15

−1

/µ

( − 0.2 ± 1.2) ×10

−15

−1

/µ

(3 ± 3) ×10

−14

−1

The numerical value of the Rydberg constant, from

the point of view of fundamental physics, can be ex-

pressed in terms of the caesium hyperﬁne interval in

atomic units and its variation may be expressed in terms

of the variations of α and µ

/µ

. A constraint for the

latter is presented in Table 30.4.

30.3.2 Constraints from Microwave Clocks

A model-independent comparison of different HFS

transitions is not simple because their nonrelativistic

contributions f

are not the same, but involve differ-

ent magnetic moments. Applying (30.9) to experimental

data, one can obtain constraints on the relative variations

of the magnetic moments of Rb, Cs, and Yb (Table 30.4).

–10

–20

–4 –3 –2 –1 0 1 2

d ln f/dt(10

–15

–1

)

Fig. 30.4 Frequency variations versus their sensitivity κ.

–5

–10

–4 –2 0 2 4

δ ln Ry/δt(10

–15

–1

)

δ ln α/δt(10

–15

–1

)

Fig. 30.5 Constraints on the time variations of the ﬁne

structure constant α and the numerical value of the Rydberg

constant. The preliminary data on Ca are not included

Part B 30.3

462 Part B Atoms

30.3.3 Model-Dependent Constraints

In order to gain information on constants more funda-

mental than the nuclear magnetic moments, any further

evaluation of the experimental data should involve the

Schmidt model, which is far from perfect. Model-

dependent constraints are summarized in Table 30.5.

The nucleon g factors, in their turn, depend on a di-

mensionless fundamental constant m

/Λ

QCD

,wherem

is the quark mass and Λ

QCD

is the quantum chro-

modynamic (QCD) scale. A study of this dependence

may supply us with deep insight into the possible vari-

ations of the more fundamental properties of Nature

Table 30.5 Model-dependent laboratory constraints on

possible time variations of fundamental constants. The

uncertainties here do not include uncertainties from the

application of the Schmidt model

X ∂ ln X/∂t

(2.9 ± 6.2) ×10

−15

−1

/µ

(2.9 ± 5.8) ×10

−15

−1

( − 0.1 ± 0.5) ×10

−15

−1

(3 ± 3) ×10

−14

−1

(see [30.31,32] for details). This approach is promising,

but its accuracy needs to be better understood.

30.4 Summary

The results collected in Tables 30.4 and 30.5 are com-

petitive with data from other searches and have a more

reliable interpretation. The results from astrophysical

searches and the study of the samarium resonance from

Oklo data claim higher sensitivity (see, e.g., [30.4]),

however, they are more difﬁcult to interpret. We have,

for example, not assumed any hierarchy in variation rates

or that some constants stay ﬁxed while others vary, as it is

done in the study of the position of the Oklo resonance.

The evaluation presented here is transparent, and any

particular calculation or measurement can be checked. In

contrast, the astrophysical data show signiﬁcant results

only after an intensive statistical evaluation.

The laboratory searches involving atomic clocks

have deﬁnitely shown progress and in a few years we

expect an increase in the accuracy of these clocks,

an increase in the number of different kinds of fre-

quency standards (e.g., optical Sr, Sr

,In

standards

and a microwave Hg

standard are being tried now),

and indeed an increase in the time separation between

accurate experiments, since it is now typically only

2–3 years. An optical clock based on a nuclear transi-

tion in Th-229 is also under consideration [30.38]. Such

a clock would offer different sensitivity to systematic ef-

fects, as well as to variations of different fundamental

constants.

Laboratory searches are not necessarily limited to

experiments with metrological accuracy. An example of

a high-sensitivity search with a relatively low accuracy is

the study of the dysprosium atom for a determination of

the splitting between the 4f

5d 6s and 4f

6s states,

which offers a great sensitivity value of κ  5.7×10

[30.36].

Variations of constants on the cosmological time

scale can be expected but the magnitude, as well as other

details, is unclear. Because of a broad range of options

there is a need for the development of as many different

searches as possible, and the laboratory search for vari-

ations is an attractive opportunity to open up a way that

could lead to new physics.

References

30.1 W. E. Baylis, G. W. F. Drake: Chapt. 1 in this Hand-

book.

30.2 P. J. Mohr, B. N. Taylor: Rev. Mod. Phys. 77, 1 (2005)

30.3 S. G. Karshenboim: Time and space variation of

Fundamental constants: Motivation and laboratory

search. Eprints physics/0306180;

S. G. Karshenboim: Search for possible vari-

ation of the ﬁne structure constant. Eprints

physics/0311080. To be published.

30.4 S. G. Karshenboim, E. Peik (Eds.): Astrophysics,

Clocks and Fundamental Constants, Lect. Notes

in Phys., Vol. 648 (Springer, Berlin, Heidelberg

2004)

30.5 N. F. Ramsey: Rev. Mod. Phys. 62, 541 (1990)

30.6 A. Bauch, H. R. Telle: Rep. Prog. Phys. 65, 789 (2002)

30.7 H. Marion, F. Pereira Dos Santos, M. Abgrall,

S. Zhang, Y. Sortais, S. Bize, I. Maksimovic,

D. Calonico, J. Gruenert, C. Mandache, P. Lemonde,

G. Santarelli, Ph. Laurent, A. Clairon, C. Salomon:

Phys. Rev. Lett. 90, 150801 (2003)

30.8 W. Paul: Rev. Mod. Phys. 62, 531 (1990) See also:

J. Javanainen, Chapt. 75 in this Handbook.

Part B 30

Atomic Clocks and Constraints on Variations of Fundamental Constants References 463

30.9 H. Dehmelt: IEEE Trans. Instrum. Meas. 31,83

(1982)

30.10 J. Stenger, C. Tamm, N. Haverkamp, S. Weyers,

H. R. Telle: Opt. Lett. 26, 1589 (2001)

30.11 T. Udem, S. A. Diddams, K. R. Vogel, C. W. Oates,

E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger,

J. C. Bergquist, L. Hollberg: Phys. Rev. Lett. 86,

4996 (2001)

30.12 S. Bize, S. A. Diddams, U. Tanaka, C. E. Tan-

ner, W. H. Oskay, R. E. Drullinger, T. E. Parker,

T. P. Heavner, S. R. Jefferts, L. Hollberg, W. M. Itano,

D. J. Wineland, J. C. Bergquist: Phys. Rev. Lett. 90,

150802 (2003)

30.13 P. T. H. Fisk, M. J. Sellars, M. A. Lawn, C. Coles: IEEE

Trans. UFFC 44, 344 (1997)

30.14 P. T. Fisk: Rep. Prog. Phys. 60, 761 (1997)

30.15 R. B. Warrington, P. T. H. Fisk, M. J. Wouters,

M. A. Lawn: Proceedings of the 6th Symposium Fre-

quency Standards and Metrology (World Scientiﬁc,

Singapore 2002) p. 297

30.16 C. J. Bordé: Phys. Lett. A 140, 10 (1989)

30.17 G. Wilpers, T. Binnewies, C. Degenhardt, U. Sterr,

J. Helmcke, F. Riehle: Phys. Rev. Lett. 89, 230801

(2002)

30.18 F. Riehle, C. Degenhardt, Ch. Lisdat, G. Wilpers,

H. Schnatz, T. Binnewies, H. Stoehr, U. Sterr: An op-

tical frequency standard with cold and ultra-cold

calcium atoms. In: Astrophysics, Clocks and Fun-

damental Constants, Lect. Notes in Phys., Vol. 648,

ed. by S. G. Karshenboim, E. Peik (Springer, Berlin,

Heidelberg 2004) p. 229

30.19 H. Katori, M. Takamoto, V. G. Pal’chikov, V. D. Ovsian-

nikov: Phys. Rev. Lett. 91, 173005 (2003)

30.20 M. Niering, R. Holzwarth, J. Reichert, P. Pokasov,

T. Udem, M. Weitz, T. W. Hänsch, P. Lemonde,

G. Santarelli, M. Abgrall, P. Laurent, C. Sa-

lomon, A. Clairon: Phys. Rev. Lett. 84,5496

(2000)

30.21 M. Fischer, N. Kolachevsky, M. Zimmermann,

R. Holzwarth, Th. Udem, T. W. Hänsch, M. Ab-

grall, J. Grünert, I. Maksimovic, S. Bize, H. Marion,

F. Pereira Dos Santos, P. Lemonde, G. Santarelli,

P. Laurent, A. Clairon, C. Salomon, M. Haas,

U. D. Jentschura, C. H. Keitel: Phys. Rev. Lett. 92,

230802 (2004)

30.22 T. Udem, J. Reichert, R. Holzwarth, S. Diddams,

D. Jones, J. Ye, S. Cundiff, T. W. Hänsch, J. Hall:

A new type of frequency chain and its application

to fundamental frequency metrology. In: The Hy-

drogen Atom: Precision Physics of Simple Atomic

Systems, Lect. Notes in Phys., Vol. 570, ed. by

S. G. Karshenboim, F. S. Pavone, F. Bassani, M. In-

guscio, T. W. Hänsch (Springer, Berlin, Heidelberg

2001) p. 229

30.23 E. Peik, B. Lipphardt, H. Schnatz, T. Schneider,

Chr. Tamm, S. G. Karshenboim: Phys. Rev. Lett.

93, 170801 (2004)

30.24 J. Sapirstein, Chapt. 27 in this Handbook

30.25 P. Mohr, Chapt. 28 in this Handbook

30.26 M. P. Savedoff: Nature 178, 688 (1956)

30.27 R. I. Thompson: Astrophys. Lett. 16,3(1975)

30.28 D. A. Varshalovich, A. V. Ivanchik, A. V. Orlov,

A. Y. Potekhin, P. Petitjean: Current status of the

problem of cosmological variability of fundamental

physical constants. In: Precision Physics of Sim-

ple Atomic Systems, Lect. Notes in Phys., Vol. 627,

ed. by S. G. Karshenboim, V. B. Smirnov (Springer,

Berlin, Heidelberg 2003) p. 199

30.29 S. G. Karshenboim: Can. J. Phys. 78, 639 (2000)

30.30 J. D. Prestage, R. L. Tjoelker, L. Maleki: Phys. Rev.

Lett. 74, 3511 (1995)

30.31 V. V. Flambaum: Limits on temporal variation

of ﬁne structure constant, quark masses and

strong interaction from atomic clock experiments.

In: Laser Spectroscopy, ed. by P. Hannaford,

A. Sidorov, H. Bachor, K. Baldwin (World Scientiﬁc,

New Jersey, London, Singapore, Shanghai, Taipei,

Chennai 2004) p. 49

30.32 V. V. Flambaum, L. B. Leinweber, A. W. Thomas,

R. D. Young: Phys. Rev. 69, 115006 (2004)

30.33 V. A. Dzuba, V. V. Flambaum, J. K. Webb: Phys. Rev.

Lett. 82, 888 (1999)

30.34 V. A. Dzuba, V. V. Flambaum, J. K. Webb: Phys. Rev.

A 59, 230 (1999)

30.35 V. A. Dzuba, V. V. Flambaum: Phys. Rev. A 61,

034502 (2001)

30.36 V. A. Dzuba, V. V. Flambaum, M. V. Marchenko:

Phys. Rev. A 68, 022506 (2003)

30.37 T. J. Quinn: Metrologia 40, 103 (2003)

30.38 E. Peik, Chr. Tamm: Europhys. Lett. 61, 181 (2003)

Part B 30

465

Molecules

Part C

Part C Molecules

31 Molecular Structure

David R. Yarkony, Baltimore, USA

32 Molecular Symmetry and Dynamics

William G. Harter, Fayetteville, USA

33 Radiative Transition Probabilities

David L. Huestis, Menlo Park, USA

34 Molecular Photodissociation

Abigail J. Dobbyn, Göttingen, Germany

David H. Mordaunt, Göttingen, Germany

Reinhard Schinke, Göttingen, Germany

35 Time-Resolved Molecular Dynamics

Volker Engel, Würzburg, Germany

36 Nonreactive Scattering

David R. Flower, Durham, United Kingdom

37 Gas Phase Reactions

Eric Herbst, Columbus, USA

38 Gas Phase Ionic Reactions

Nigel G. Adams, Athens, USA

39 Clusters

Mary L. Mandich, Murray Hill, USA

40 Infrared Spectroscopy

Henry Buijs, Québec, Canada

41 Laser Spectroscopy in the Submillimeter

and Far-Infrared Regions

Kenneth M. Evenson

†

John M. Brown, Oxford, England

42 Spectroscopic Techniques: Lasers

Paul Engelking, Eugene, USA

43 Spectroscopic Techniques:

Cavity-Enhanced Methods

Barbara A. Paldus, Palo Alto, USA

Alexander A. Kachanov, Sunnyvale, USA

44 Spectroscopic Techniques: Ultraviolet

Glenn Stark, Wellesley, USA

Peter L. Smith, Cambridge, USA

467

Molecular Stru

31. Molecular Structure

Molecular structure is a reﬂection of the Born–

Oppenheimer separation of electronic and

nuclear motion, which is in turn a consequence

of the large difference between the electron

and nuclear masses. One consequence of this

separation is the concept of a potential energy

surface for nuclear motion created by the faster

moving electrons. Corollaries include equilibrium

structures, transition states, and reaction paths

which are the foundation of the description

of molecular structure and reactivity. However

the Born–Oppenheimer approximation is not

uniformly applicable and its breakdown results in

perturbations in molecular spectra, radiationless

decay, and nonadiabatic chemical reactions.

There are many issues that can be addressed

in a discussion of molecular structure, including

the structure and bonding of individual classes

of molecules, computational and/or experimental

techniques used to determine or infer molecular

structure, the accuracy of those methods, etc.

In an effort to provide a broad view of the

essential aspects of molecular structure, this

Chapter considers issues in molecular structure

from a theoretical/computational perspective

using the Born–Oppenheimer approximation

as the point of origin. Rather than providing

a compendium of results, this chapter will explain

how issues in molecular structure are investigated

and how the questions that can be addressed

reﬂect the available methodology. Even with

these restrictions the scope of this topic remains

enormous and precludes a detailed presentation

of any one issue. Thus the abbreviated discussions

in this work are supplemented by ample references

to the literature.

Several aspects of potential energy surfaces

and their relation to molecular structure will be

considered: (i) the electronic structure techniques

used to determine a single point on a potential

energy surface, and the interactions that couple

the electronic states in question, (ii) the local

properties of potential energy surfaces, in

particular equilibrium structures and rovibrational

31.1 Concepts ............................................. 468

31.1.1 Nonadiabatic Ansatz:

Born–Oppenheimer

Approximation.......................... 468

31.1.2 Born–Oppenheimer Potential

Energy Surfaces

and Their Topology .................... 469

31.1.3 Classiﬁcation of Interstate

Couplings: Adiabatic

and Diabatic Bases .................... 469

31.1.4 Surfaces of Intersection

of Potential Energy Surfaces ....... 470

31.2 Characterization of Potential

Energy Surfaces ................................... 470

31.2.1 The Self-Consistent Field (SCF)

Method.................................... 471

31.2.2 Electron Correlation:

Wave Function Based Methods.... 472

31.2.3 Electron Correlation:

Density Functional Theory .......... 475

31.2.4 Weakly Interacting Systems ........ 476

31.3 Intersurface Interactions: Perturbations 476

31.3.1 Derivative Couplings .................. 476

31.3.2 Breit–Pauli Interactions ............. 477

31.3.3 Surfaces of Intersection.............. 479

31.4 Nuclear Motion.................................... 480

31.4.1 General Considerations .............. 480

31.4.2 Rotational-Vibrational Structure . 481

31.4.3 Coupling of Electronic

and Rotational Angular

Momentum in Weakly Interacting 482

31.4.4 Reaction Path ........................... 483

31.5 Reaction Mechanisms:

A Spin-Forbidden Chemical Reaction ..... 484

31.6 Recent Developments........................... 486

References .................................................. 486

levels that provide the link to experimental

inferences concerning molecular structure,

(iii) global chemistry deduced from potential

energy surfaces including reaction mechanisms

and reaction paths and (iv) phenomena resulting

from the nonadiabatic interactions that couple

potential energy surfaces.

Part C 31

468 Part C Molecules

31.1 Concepts

31.1.1 Nonadiabatic Ansatz:

Born–Oppenheimer Approximation

Basic Quantities

The total Hamiltonian for electronic and nuclear motion

in the space ﬁxed coordinate frame, in atomic units, is

(r, R) =



α=1

−1

∇

+ H

(r; R)

≡ T

nuc

+ H

(r; R), (31.1)

where R denotes the 3N nuclear coordinates, with R=

, R

,... ,R

), R

= (X

, Y

, Z

), r denotes the 3M

electronic coordinates 2using similar conventions, T

nuc

is the nuclear kinetic energy operator and H

(r, R) is

the total electronic Hamiltonian taken as

(r; R) = H

(r; R) + H

rel

(r; R). (31.2)

Here H

(r; R) is the nonrelativistic Born–Oppenheimer

Hamiltonian

(r; R) =−



i=1

∇

−



K,i

− r



i= j

− r



K=L

− R

(31.3)

and H

rel

(r; R) is the relativistic contribution to the

electronic Hamiltonian, for light atoms conventionally

treated within the Breit–Pauli approximation [31.1]and

discussed further in Sect. 31.3.2. Note that in (31.3)the

nuclear kinetic energy is absent, so that only the r

are dynamical variables in (31.2)and(31.3). Using the

Born–Huang approach [31.2] to the Born–Oppenheimer

approximation, the total wave function Ψ

(r, R) is ex-

panded in a basis of electronic states, Ψ

(r; R). The total

wave function for the system thus has the form

(r, R) =



(r; R)β

(R). (31.4)

Equation (31.4) is valid for any complete set of

electronic states depending parametrically on nuclear

coordinates. The parametrical dependence of Ψ

(r; R)

on R, denoted by the semicolon, reﬂects the fact that

the R are not dynamical variables in (31.2)and(31.3).

As a practical matter it is necessary to make a particular

choice of Ψ

(r; R) in order to limit the size of the expan-

sion in electronic states. An adiabatic electronic state is

an eigenfunction of H

(r; R) for ﬁxed R. A particularly

useful choice of adiabatic state, denoted by Ψ

(r; R),

employs H

(r; R) rather than the full H

(r, R).These

electronic wave functions satisfy

(r; R)Ψ

(r; R) = E

(R)Ψ

(r; R). (31.5)

The Ψ

(r; R) are determined up to a geometry de-

pendent phase. This phase is usually chosen such that

the Ψ

(r; R) are real. This assumption is acceptable

except in the situation where there is a conical inter-

section on the potential energy surface in question. In

that case, the real-valued Ψ

(r; R) changes sign when

a closed loop surrounding the conical intersection point

is traversed, that is, Ψ

(r; R) is not single-valued [31.3].

This geometric or Berry [31.4] phase condition has

consequences in such phenomena as the dynamic Jahn–

Teller effect [31.5,6] but will not be addressed in detail

in this review.

As a consequence of the parametric dependence of

(r; R) on R, E

(R) becomes a function of R and

is referred to as the nonrelativistic Born–Oppenheimer

potential energy surface for reasons discussed below.

Approaches based on (31.5) are most appropriate for

molecular systems with only light atoms. However with

the use of pseudopotential techniques [31.7], formally

equivalent approaches can be developed for heavier

systems (Sect. 31.3.2).

Inserting (31.4) into the time independent Schrö-

dinger equation



(r; R) − E



(r; R) = 0and

taking the inner product with the electronic basis state

(r; R) gives the following system of coupled equa-

tions for the rovibronic functions β

(R) [31.8,9]:



nuc

+ E

(R) − E



(R)

=−





(R) + H

(R)

−



α=1





(R) · ∇

+∇

· f

(R)







×β

(R), (31.6)

where the state label L on β

has been suppressed, and

(R) ≡



(R), f

(R)



, (31.7)

(R) =



W,α

(R), (31.8)

(R) =



(r; R)



∂

∂W

(r; R)



, (31.9)

Part C 31.1

Molecular Structure 31.1 Concepts 469





∂

∂W

(r; R)



∂

∂W



(r; R)



(31.10)

with W = X, Y, Z,α= 1 ···N. The subscript r on the

matrix elements in (31.9–31.10) denotes integration over

all electronic coordinates and the ﬁnal ∇

in (31.6)

acts on both f

(R) and β

(R). While this represen-

tation may not be optimal for a treatment of the nuclear

dynamics, since it employs a space ﬁxed frame repre-

sentation, it is adequate for the present development as

it clearly displays the origins of the Born–Oppenheimer

approximation and its breakdown.

Modiﬁcations to the Nonrelativistic

Born–Oppenheimer Potential Energy Surface

When the interstate couplings on the right-hand side

of (31.6) can be neglected, the nuclear motion is gov-

erned by the effective potential given by V

(R) ≡

(R)+

(R) ≡ E

(R) + H

(R) +

(R) and the

Born–Oppenheimer approximation is valid. Here it is as-

sumed that f

(R) = 0, a sufﬁcient condition for which

is that the Ψ

(r; R) are chosen real-valued [see discus-

sion following (31.5)]. E

(R) is generally the principal

R-dependent contribution to V

(R). H

(R) is referred

to as the relativistic contribution to the nonrelativistic

Born–Oppenheimer potential energy surface and is dis-

cussed further in Sect. 31.3.2. The contribution

(R)

is referred to as the adiabatic correction. Unlike E

(R)

and H

(R), which are independent of mass,

(R)

is mass dependent (31.8 and 31.10). It has been com-

puted from ﬁrst principles [31.10–12] and inferred from

experiments [31.13, 14] principally for diatomic sys-

tems.

31.1.2 Born–Oppenheimer Potential Energy

Surfaces and Their Topology

(R) is the main focus of this chapter. There have been

several recent reviews of ground state (I = 1) potential

energy surfaces [31.15,16].

Equilibrium structures (stable or metastable species)

represent local minima on E

(R),thatisg

(R) ≡

∂ E

(R)/∂W

= 0forallW

at R = R

.Theremay

be several equilibrium structures for a given set of nu-

clei; for example the atoms H, C and N form stable

molecules HCN and HNC. Saddle points or transition

states, extrema on E

(R) with one negativeeigenvalue of

the Hessian matrix F

(R), whose elements are deﬁned

by F

(R) ≡ ∂

(R)/∂W

∂W

, represent moun-

tain passes separating the various equilibrium structures

and the asymptotes – values ofR corresponding to iso-

lated molecular fragments. This situation is illustrated

in Fig. 31.1.

Reaction paths [31.17] connect the asymptotes, min-

ima, and saddle points. They can best be deﬁned as

the steepest descent paths from a transition state struc-

ture down to local minima [31.17], although methods

for walking uphill (shallowest ascent path) [31.18]

also exist. Unlike equilibrium structures and saddle

points which are independent of the choice of coor-

dinate system used to representR, the reaction path

is coordinate system dependent. The intrinsic reac-

tion coordinate [31.17], the reaction coordinate in mass

weighted cartesian coordinates R

→ R

, is most fre-

quently used (Sect. 31.4.4)

These features of the potential energy surfaces and

their determination will be discussed further below.

Their determination consists of two parts: (i) the level

of treatment, i. e., the electronic structure technique

used to determine E

(R) and (ii) the characteriza-

tion of the features of E

(R), i. e., minima, saddle

points, and reaction paths [31.19,20] at the level of treat-

ment chosen. In this chapter, only fully ab initio levels

of treatment are considered [31.21,22].

31.1.3 Classiﬁcation of Interstate Couplings:

Adiabatic and Diabatic Bases

Intersurface couplings, shown on the right-hand side

of (31.6), result in the breakdown of the single poten-

tial energy surface Born–Oppenheimer approximation.

Within the Born–Huang ansatz such a nonadiabatic pro-

cess is interpreted in terms of motion on more than

one Born–Oppenheimer potential energy surface [31.8].

From (31.6), it is seen that there are three types of ma-

trix elements coupling the electronic states,

(R),

(R) and f

(R). Two of these couplings,

(R)

Transition state

Equilibrium

structure

Reaction path

Equilibrium

structure

Fig. 31.1 Schematic representation of a transition state and

reaction path connecting two minima

Part C 31.1

470 Part C Molecules

and f

(R) arise from the nuclear kinetic energy op-

erator T

nuc

, while the third coupling arises from the

Breit–Pauli interaction H

. This classiﬁcation of the

intersurface interactions is a consequence of the choice

of the adiabatic electronic states through (31.5). Other

choices of Ψ

(r; R), most notably the diabatic electronic

states [31.23–27], are possible.

In the adiabatic state basis, H

(r; R) is diagonal and

in the absence of relativistic effects, intersurface coup-

lings originate exclusively from the derivative coupling

terms in (31.9)and(31.10). The diabatic basis seeks to

‘transfer’ the intersurface coupling from these deriva-

tive operators to a potential term analogous in form to

(R). The diabatic basis, Ψ

0−d

(r; R), is a unitary

transformation of the adiabatic electronic basis, deﬁned

such that [31.24]

JI,d

≡



0−d

(r; R)



∂

∂R

0−d

(r; R)



= 0

(31.11)

where R

is an internal coordinate. For polyatomic

systems (N > 2), rigorous diabatic bases do not ex-

ist [31.27] and approximate diabatic bases are sought

[31.27–30]. A discussion of this issue can be found

in [31.31].

31.1.4 Surfaces of Intersection of Potential

Energy Surfaces

The intersurface couplings are most effective in promot-

ing a nonadiabatic process when |V

(R) − V

(R)| is

small for some J on the right-hand side of (31.6), so

that

(R) +

(R) −



(R) +

(R)



≡ ∆E

(R) + ∆E

(R) + ∆K

(R) (31.12)

is small. Here

∆E

(R) ≡ E

(R) − E

(R); , (31.13)

∆K

(R) ≡

(R) −

(R), (31.14)

∆E

(R) ≡ H

(R) − H

(R). (31.15)

Since in general ∆K

(R) is quite small, and for the

low atomic number systems considered here H

is it-

self small, we are led to consider regions of nuclear

coordinate space for which the magnitude of ∆E

(R)

is small, referred to as avoided intersections when

∆E

(R) = 0 and as conical and Renner-type [31.32]

intersections when ∆E

(R) = 0. The set of Rfor which

∆E

(R) = 0, the noncrossing rule, was ﬁrst discussed

in 1929 by von Neumann and Wigner [31.33]. The rule

states that for diatomic systems, crossings between two

potential energy curves of the same symmetry are not

possible (actually, are extremely rare), whereas for poly-

atomic systems, potential energy surface intersections

are allowed. Mathematically the noncrossing rule is ex-

pressed in terms of the dimensions of a surface, the

surface of intersection, on which a set of conditions

can be satisﬁed [31.3, 34]. For electronic states of the

same symmetry (neglecting electron spin degeneracy),

the dimension of the surface of conical intersection is

K − 2, where K is the number of internal nuclear de-

grees of freedom. For states of different spin symmetry,

the dimension of the surface of intersection is K − 1.

31.2 Characterization of Potential Energy Surfaces

This section is concerned with the determination of the

approximate solution of (31.5). For this purpose it is

frequently convenient to re-express H

(r; R) in second

quantized [31.35]formas

(r; R) =



i, j

†



i, j,k,l

(il| jk)a

†

(31.16)

where h

and (il| jk) are standard abbreviations [31.36]

for the one electron (kinetic energy and nuclear–electron

attraction) and two electron (electron–electron repul-

sion) integrals respectively, and a

†

and a

are the

fermion creation and destruction operators. The inte-

grals and the creation and destruction operators are

deﬁned in terms of a basis of one electron func-

tions, usually molecular spin-orbitals φ

, γ

= α or β.

Thus

(R) =



)γ



−

∇

−



− r



)γ



(31.17)

(ij|kl) =



)γ



− r



)γ



. (31.18)

Part C 31.2

Molecular Structure 31.2 Characterization of Potential Energy Surfaces 471

Equation (31.18) is referred to as the Mulliken no-

tation [31.37] for a two-electron integral. It differs

from the frequently used [31.36] bra-ket representa-

tion ij|kl=(ik| jl). The molecular orbitals φ

; R)

are in turn expanded in terms of a basis set

of atom-centered functions [31.38], χ(r

; R), with

; R) =



P=1

; R)T

P,i

(R). Note that here we

have distinguished between a spin-orbital φ

; R)γ

and a molecular orbital φ

; R), but we will fol-

low the usual convention of allowing the functions

in (31.17)and(31.18) to be molecular orbitals

or spin-orbitals with the use being clear from the

context.

Since H

(r; R) is independent of electron spin, it

can also be written in terms of the

, spin-averaged

excitation operators [31.39], where



†

pγ

qγ

p, q label orbitals and γ = α, β so that

(r; R) =



i, j



i, j,k,l

(ij|kl)

− δ

(31.19)

The

satisfy

−

as a conse-

quence of the commutation relations for the a

†

and a

and

are generators of the unitary group [31.40], (Chapt. 4).

This observation allows the powerful machinery of the

unitary group to be applied to the evaluation of E

(R)

[31.41] (Sect. 31.1.2).

The determination of E

(R) involves two steps;

(i) the determination of the molecular orbitals φ

; R),

and (ii) sometimes simultaneously, the determination of

the approximate wave function Ψ

(r; R).

Wave functions discussed in this section are

constructed from functions that are antisymmetrized

products of spin-orbitals φ

; R)γ

(r; R) = A



i=1

; R)γ

, (31.20)

where γ

= α or β. The antisymmetrizer A is required

to take into account the Fermi statistics of the elec-

trons. The wave function in (31.20)isreferredtoas

a Slater determinant and is an eigenfunction of M

,the

z-component of total electron spin. A linear combina-

tion of Slater determinants that is also an eigenfunction

of S

is referred to as a conﬁguration state func-

tion (CSF) [31.42]. Equation (31.20) corresponds to

a particular distribution of electrons in the orbitals

; R). This distribution is referred to as an electron

conﬁguration.

In principle, it is possible to learn everything about

the topology of a potential energy surface from the

pointwise evaluation of E

(R). In practice however,

the determination of topological features, including

location of equilibrium structures, saddle points and

reaction paths, characterization of quadratic and an-

harmonic force ﬁelds [31.43],andeventheevaluation

of derivative couplings [31.44], has beneﬁtted im-

mensely from techniques in which the energy gradient

(R) or higher derivatives are determined directly

from knowledge of the wave function at the R in

question. These analytic derivative techniques, i. e.,

techniques that do not use divided difference dif-

ferentiation, have been actively developed since their

ﬁrst introduction [31.45].

The techniques most commonly used to calculate

(R) and its derivatives are described below.

31.2.1 The Self-Consistent Field (SCF)

Method

The SCF Energy

In this most basic treatment, the electronic wave

function Ψ

0−SCF

is taken as a single Slater de-

terminant or a single CSF. The spin-orbitals are

determined to give an extremum of the energy func-

tional



0−SCF

(r; R)



(r; R)



0−SCF

(r; R)



.This

gives rise to the self-consistent ﬁeld conditions. For re-

stricted closed shell wave functions [31.46], deﬁned by

the conditions

0−RSCF

(r; R)

= A



i=1

; R)αφ

2i+1

; R)β (31.21)

with φ

; R) = φ

2i−1

; R), i = 1,... ,L and 2L =

M,theSCF equations are

= h



j ∈{j

occ

}

2(ir| jj) − (ij|rj) = 0 (31.22)

for φ

, φ

occupied and virtual orbitals, respectively.

Here, occupied orbitals j

occ

are those occurring in

the orbital product in (31.21), and the remainder of

the orbitals are referred to as virtual orbitals. The

solution to (31.22) is referred to as the canonical

SCF solution, provided the orbitals satisfy the SCF

equations

Fφ

= λ

, (31.23)

Part C 31.2

472 Part C Molecules

where the Fock operator F is deﬁned by F

= ε

and λ

= ε

. The corresponding electronic energy is

(R) ≡ E

0−SCF

(R) =



j ∈{j

occ

}

(ε

+ h

SCF Energy Derivatives

The derivative of E

0−SCF

(R) can be expanded through

third order using analytic gradient techniques [31.47,

48], and through fourth order using divided differ-

ences and more recently using analytic derivative

techniques [31.43]. The ability to evaluate the energy

derivatives or force ﬁeld through fourth order is im-

portant in the determination of vibrational properties of

molecules, as discussed in Sect. 31.4.2.

Direct SCF

The range of molecular systems accessible to treatment

at the SCF level has been expanded considerably by

the introduction of the direct SCF methods [31.49]. In

this method, none of the two-electron integrals in the

χ basis (pq|rs) are stored during the iterative solution of

(31.22)and(31.23). Since the number of such integrals

grows as L

/8, direct SCF procedures avoid the ‘L

storage bottleneck’, enabling SCF wave functions to be

determined for systems with more than 100 atoms and

basis sets with more than 1000 functions. (If all the L

integrals were stored on disk this basis would require

≈ 1000 gigabytes).

31.2.2 Electron Correlation:

Wave Function Based Methods

Wave functions more accurate than SCF wave func-

tions are obtained by including the effects of electron

correlation. The correlation energy is deﬁned as

corr

(R) ≡ E

(R) − E

0−SCF

(R). Methods for the deter-

mination of E

corr

(R) are commonly classiﬁed as single

reference or mulitreference methods. In single reference

methods, an SCF wave function Ψ

0−SCF

(r; R) given

by (31.20) is improved. In multireference methods, the

starting point is the space spanned by a set of terms

like (31.20). This space is referred to as the reference

space and the functions in the space as the reference con-

ﬁgurations. Sometimes the molecular orbitals φ(r

; R)

for the reference space are the SCF orbitals of a sin-

gle reference conﬁguration calculation, but more often

the φ(r

; R) are chosen to satisfy multiconﬁgurational

self-consistent ﬁeld (MCSCF) equations [31.50–53].

Single Reference Methods

Second Order Møller–Plesset Perturbation Theory

(MP2).

In this approach [31.54], the solutions of the SCF

equations are used to determine E

(R) to second order

0−MP2

(R) = E

0−SCF

(R)

−



a,b,i, j

(ab||ij )

+ λ

− λ

(31.24)

where i, j denote occupied orbitals, a, b denote virtual

orbitals, and

(ab||ij ) = (ai|bj) − (aj|bi) =ab|ij−ab| ji .

Both gradients of E

0−MP2

(R) and direct implementa-

tions of MP2 [31.37,55] are currently available. Higher

order perturbation theories, also in common use, are dis-

cussed in the context of coupled cluster methods which

are considered next.

Coupled Cluster Method. Perhaps the most reliable

method currently available for characterizing near equi-

librium properties of moderately sized molecules is the

coupled cluster approach [31.56–60](Chapt.5). In this

approach, the exact wave function is written as a unitary

transformation of a reference wave function Φ

(r; R)



usually, but not necessarily, the SCF wave function

(r; R) = Ψ

0−SCF

(r; R)



0−CC

= exp(T )Φ

. (31.25)

where T is an excitation operator deﬁned as

T =



; (31.26)

with



i, j,k,... ,a,b,c

abc···

ijk···

†

, (31.27)

where i, j, k denote occupied orbitals in Φ

and a, b, c

denote virtual orbitals. Thus

0−CC

= Φ



i,a



i, j,a,b

+··· ,

(31.28)

where Φ

, Φ

are single and double excitations

from Φ

, and it is the amplitudes t that must be de-

termined. Wave functions of the form in (31.25–31.28)

have the important property referred to as size con-

sistency [31.60] or size extensivity [31.59]. Consider

N noninteracting helium atoms. The electronic wave

Part C 31.2