MODERN ELECTRONIC STRUCTURE THEORY › ... › 82_ModernElectronicStructureTheory_19… · P* cm....

MODERN ELECTRONIC STRUCTURE THEORY Part I

Editor

David R Yarkony Department of Chemistry- Johns Hopkins University USA

b World Scientific b s* lngapore l New Jersey l London * Hong Kong

P u b l i s h e d b y

W o r l d S c i e n t i f i c P u b l i s h i n g C o . P t e . L t d .

P 0 B o x 1 2 8 , F a r r e r R o a d , S i n g a p o r e 9 1 2 8

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1 . A t o m i c t h e o r y . I . Y a r k o n y , D a v i d . I I . S e r i e s . Q D 4 6 1 . M 5 9 6 7 1 9 9 5 5 4 1 . 2 ’ 8 - - d c 2 0 9 4 - 4 3 3 8 5

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A l l r i g h t s r e s e r v e d . T h i s b o o k , o r p a r t s t h e r e o J m a y n o t b e r e p r o d u c e d i n a n y f o r m o r b y a n y m e a n s , e l e c t r o n i c o r m e c h a n i c a l , i n c l u d i n g p h o t o c o p y i n g , r e c o r d i n g o r a n y i n f o r m a t i o n s t o r a g e a n d r e t r i e v a l s y s t e m n o w k n o w n o r t o b e i n v e n t e d , w i t h o u t w r i t t e n p e r m i s s i o n f r o m t h e P u b l i s h e r .

F o r p h o t o c o p y i n g o f m a t e r i a l i n t h i s v o l u m e , p l e a s e p a y a c o p y i n g f e e t h r o u g h t h e C o p y r i g h t C l e a r a n c e C e n t e r , I n c . , 2 2 2 R o s e w o o d D r i v e , D a n v e r s , M a s s a c h u s e t t s 0 1 9 2 3 , U S A .

P r i n t e d i n S i n g a p o r e b y U t e P r i n t

CHA ER 1

EXACT EXPANSION METHO S FOR ATOMIC HYDROGEN

IN AN EXTERNAL ELECTROSTATIC FIELD: DIVERGENT

PERTURBATION SERIES, BOREL SUMMABILITY,

SEMICLASSICAL APPROXIMATION, AND EXPANSION

OF PHOTOIONIZATION CROSS-SECTION

OVER RESONANCE EIGENVALUES

Harris J. Silverstone

Department of Chemistry, The Johns Hopkins University,

Baltimore, Maryland 21218-2685, USA

Contents

I. Introduction 2. Formal Rayleigh-Schrodinger Perturbation Theory

2.1. Convergence: Three Examples of RSPT for the Barmonic Oscillator 2.1.1. Trivial Perturbation: Linear Potential 2.1.2. Simple Perturbation: Change in Force Constant 2.1.3. Anharmonic Perturbation: Divergent RSPT

3. Power Series 3.1. Familiar Convergent Series; Convergence in a Circle 3.2. One Familiar Divergent Series: Stirling’s Approximation 3.3. Standard Prototypical Asymptotic Power Series:

Poincare’s Definition 3.4. Bore1 Sum of an Asymptotic Power Series 3.5. Complex Bore1 Sum of a Real Asymptotic Power Series;

Stokes Line 3.6. Dispersion Relation and Asymptotic Power-Series Coefficients

4. Hydrogen Atom in Constant Electric Field 4.1. LoSurdo-Stark Effect

590

592

593

593

594

594

595

595

596

596

597

599

602 603 603

589

590 Modern _?i!~ech’onic f?h-ucb-e Theory

4.2. RSPT Following Schrodinger’s 1926 Treatment 4.2.1. Parabolic Coordinates 4.2.2. Algebraic Relations to Get E(F) from /3r(f) and &(f)

4.3. Closed Form Term-by-Term Solvability of the erturbation Series 4.4. Recursive Solution of the Perturbation Series to High Order 4.5. Divergence; Complex Bore1 Sum 4.6, Field-Ionization and Complex Energy Eigenvalues 4.7. Variational Calculation of Complex Energy Eigenvalues 4.8. Perturbation Theory for the Imaginary Terms 4.9. Dispersion Relation and Asymptotics of RSPT

5. Photoionization of Atomic Hydrogen 5.1. Complex Energies Are Not Enough 5.2. Expansion as Sum over Complex Resonances: Transition Dipoles 5.3. Comparison with Experiment

6. JWKB Connection Formulae at a Linear Turning Point 6.1. JWKB Wave Function 6.2. Bore1 Summability of the Airy Function Asymptotic Expansions 6.3. JWKB for a Linear Potential: JWKB Form for

Airy Function Asymptotic Expansions 6.4. JWKB Connection Formulae via Airy Function

Asymptotic Expansions Acknowledgments References

604 604 605 607 608 609 610 611 613 618 620 622 622 624 626 626 629

633

637 640 640

ntroduction

Schrodinger’ introduced a power series expansion to calculate the energy

levels of the hydrogen atom in an external electric field (LoSurdo-Stark

effect2T3) and in the process invented Rayleigh-Schrodinger perturbation

theory (RSPT). For the LoSurdo-Stark effect, the RSPT energy coeffi-

cients can be obtained term-by-term in closed form.4 Atomic hydrogen in

an electric field is the first example of a quantum mechanical problem that

cannot be solved exactly in closed form, but for which one can find an

infinite expansion with each term known exactly.

The RSPT series for atomic hydrogen in an electric field has an inter-

esting property. Despite the point of view that a perturbation is a small

change to the unperturbed problem, the RSPT series for the LoSurdo-

Stark effect does not converge for any nonzero value of the electric field,

no matter how small. Nevertheless, the RSPT series is Bore1 summable,5

Exact Expansion Methods for Atomic Hydrogen . . . 591

so that “sums” of the series can be found by methods other than partial

sums. The harmonic oscillator with an x4 anharmonic perturbation has an

RSPT series with similar characteristicss-lo

Hydrogen in an external electric field is a richly complicated system.

No matter how small, the external field in principle leads to tunneling-

ionization.” At low fields the ionization rate is exponentially small in the

field strength F: N ~-2nz-~+le-2/(3n3F) , where n is the principal quan-

tum number, m the magnetic quantum number, and n2 one of the parabolic

quantum numbers. “-” There are in fact no bound states, but instead com-

plex resonance wave functions and eigenvalues. The Bore1 sum of an RSPT

energy series is a complex resonance eigenvalue.5 The imaginary part of

a resonance eigenvalue gives the ionization rate of the resonance, and an

infinite expansion for the ionization rate itself can be derived in which each

term can be found exactly.15

Photoionization of hydrogen in an electric field has been studied exper-

imentally by several groups. “-” Complete theoretical description of these

experiments requires more than RSPT and more than ionization rates ob-

tained from resonance eigenvalues, because for many states the electric field

is too large for RSPT to be useful, and because it is necessary to describe

both the ionization rates and the observed asymmetrical line shapes. One

approach is to expand the photoionization cross-section as an infinite sum

over the resonances.21f22 The contribution of each term can be evaluated

exactly numerically.

Let us return to the infinite expansion for the ionization rate. A key

step is to calculate the “discontinuity” in the resonance energy eigenvalues

when the electric field becomes real from complex values above and below

the real axis. This calculation is semiclassical in nature. That is, the ap-

proach is like the Jeffrey-Wentzel-Kramers-Brillouin (JWKB) method. In

the JWKB method, the energy and the logarithm of the wave function are

expanded in power series in fi. Each term in the series can be calculated via

quadratures, if not in closed form. The JWKB series are known to be diver-

gent but asymptotic. Moreover, their explicit form changes discontinuously,

depending on whether x is in a classically allowed region or in a barrier.

The changes are what are referred to as the “JWKB connection formulae at

a linear turning point.” The discontinuity required for the LoSurdo-Stark

problem turns out to be closely related to the connection-formula problem,

and a discussion of both are warranted.

592 Modem Electronic Structure Theory

The aims of this chapter are on the one hand to examine the various as-

pects of atomic hydrogen in an electric field - the RSPT, the divergence of

the RSPT, the Bore1 summability of the RSPT, the series for the ionization

rate, the exact calculation of resonance eigenvalues, and the expansion for

the photoionization cross-section - and on the other hand, to discuss some

general aspects of the techniques - RSPT, Bore1 summability, and JWKB

connection formula problem. The unifying theme, which also accounts for

the eclectic selection of topics, is infinite expansions where the individual

terms can be found exactly.

2. Formal Rayleigh-Schrcdinger Perturbation Theory

Rayleigh-Schrodinger perturbation theory’ is the solution of the Schro-

dinger equation for a Hamiltonian as a power series in the perturbation

“parameter” A. The basic equations for the &h energy eigenstate are

straightforward to derive.

z!kIY@)+AV, (1)

,I/& = I@) + A?)?) + A2?@) + . . . , c4

En = EC) + A,?3;) + A2Ep) + . . . . (31

The power-series expansion of the Schrodinger equation,

gives the Nth order RSPT equation,

Simple formal formulae for the solution can be found for the nondegenerate

case, particularly with “intermediate normalization,”

and with the reduced-resolvent operator I&e):


(81

The solution is formal in the sense that the reduced-resolvent operator I&e)

usually is not known explicitly and hides with the symbol I&e’ the process

of solving the RSPT equations.

2.1. Conveqence: Three Examples of RSPT for

the Harmonic Oscillator

Harmonic oscillators provide work-horse test cases when the associated

equations can be solved. Here we give three harmonic-oscillator-based quick

samples of the kinds of convergence behavior that RSPT series can display:

convergence everywhere, convergence in a circle with finite radius, and di-

vergence.

The unperturbed harmonic oscillator is summarized by:

Here Hn(x) is the Hermite polynomial of degree n (Ref. 23).

2.1.1. Trivial Perturbation: Linear Potential

The linear perturbation,

AV=Ax,

is “small” compared to H(O), and in the resulting RSPT

rium position and energy-zero of the oscillator change:

ix2 + Ax = $x + A)2 - iA2,

w only the equilib-

(14)

(15)


That is, for each level,

EiNJ = 0, (N > 2).

This trivial example is mentioned only because the R§PT is valid for all

values of A.

2.1.2. Simple Perturbation: Change in Force Constant

Perturbation of the force constant is of the “same order of magnitude” as

H(O), and the result is that the frequency and energy levels scale by dm:

Every RSPT energy

binomial coefficient:

Although the energy

coefficient is nonzero and given by (n + 1/2) times a

IZiN)= (n+i) (FT. w

is given by (n + 1/2) ~,/m for all values of ?I > - 1,

SPT series for En converges only for 1 Ai < 1, and it diverges for

IAl > 1.

2.1.3. Anharmonic Perturbation: Divergent RSPT

y adding x4 anharmonicity,

which is “large” compared to II(*), the full eigenvalue problem can no longer

be solved m closed form. The RSPT equations can be solved in closed form

Exact Expansion Methods for Atomic Hydrogen . . , 595

order-by-order, which is a consequence of x 4 (*I being a linear combination $J~

Bender and Wu6 made numerically the surprising discovery in 1968 that

the RSPT series for the ground-state energy diverged. The excited-state

series diverge as well: the EL? grow factorially fast with N (Refs. 8 and 24):

Here I’(m) denotes the Gamma function: I’(m) = (m - l)!. If A is very

small, then the first few terms of the power series decrease in magnitude

as N increases. But the factorial growth eventually overwhelms the power

decrease of (3A)N, and the terms in

Does divergence mean that the

as we shall see in the next section.

ower Series

the power series increase without limit.

power series is useless? By no means

The RSPT series for a harmonic oscillator perturbed by increasingly larger

perturbations went from always convergent to always divergent. It is useful

to be able to have at one’s fingertips standard examples of convergence

behavior, the presentation of which is the task of this section.

3.1. Familiar Convewent Series; Convergence in a Circle

The geometric series has the simplest coefficients:

1 + x + x2 + . . . = l/(l - x) . cw

Successive terms are smaller, and the series converges to l/(l--x), if 1x1 < 1.

The terms get larger if 1x1 > 1, and the series diverges. In the complex x

plane, power series converge inside circles. For the geometric series, the

radius of convergence is 1. The series for dm encountered above is

similar in convergence behavior to the geometric series.

A second familiar expansion is the series for the exponential function,

l+x+ s+$-...=ex,

596 Modern Electronic Structwe Theoy

which is typical of a series with an infinite radius of convergence. No

matter what value x has, the terms eventually go strongly to 0, and the

series converges.

3.2. One Familiar Divevent Series: Stirhg’s Approximation

Not all useful series converge. Stirling’s approximation23 for N!,

x l+- C

1 1 139 571

12N + 288N2 - 51840N3 - 2488320N4 + - * * ’ “”

is one such example of a series that diverges for all N. Nevertheless, for

large N, a few terms in the series are numerically spectacularly successful.

3.3. Standard Pr&otypical Asymptotic Power Series:

Poincark’s Definition

A simpler - and consequently more suitable as a prototype - divergent

series, attributed to Euler, is

1 - x + 2!x2 - 3!x3 + - . . . _ cw

This series results from integration by parts of Juw eWt(l + xt)-‘dt, which

is a version of the exponential integral,23

JW emt(l + xi)-‘& = xV1ezW1&(x-‘) . 0

NW

One finds that

J m eBt -di? = 1 - z + 2!x2 - 3!x3 + . . .

0 1+xt

m + (n - l)!(-Lzy + f-q--q J

evt o (1 +xt)n+P* (W

The remainder term, if x is positive, is bounded by the next term in the

series.

I Jw

n!( -xy emt o cl + xtjn+l c-h! 5 T&C‘ ow evtdt = n!xn , I J

ifx>O.

Exact E x p a n s i o n Methods for Atomic Hydrogen . . . 5 9 7

For any z > 0, TZ!X~ increases without limit as n - + C G . But if z is small,

there may be values of n for which n!xn is very small. Suppose that x = 0.1.

Then n!O.ln declines to a minimum of 0.000363 at n = 9 and 10. The

partial sum through the n = 9 term is 0.915456.. . , which agrees with the

exact value 0.915633.. . to better than three digits. If more terms are kept,

the agreement worsens. In this limited-accuracy, partial-sum sense, the

divergent power series is computation&y useful.

Poincare25 captured the aspect of the competition between the small-

ness of x and the largeness of N and made it the basis of a definition of

a s y m p t o t i c power series. The series

a 0 + a l x + a 2 x 2 + . . .

is an asymptotic power series for a function f(x) if

f ( x ) - 5 a m x n = O ( [ x i N + ‘ ) n=O

as x --+ 0, where O(]X~~+~) is the usual “order” symbol that means “goes

to 0 like the quantity inside the parentheses” - in this case 1x1 N-+l. Euler’s

series clearly satisfies the Poincare definition.

The major deficiency of a divergent asymptotic series in the sense of

Poincare is that many functions have the same asymptotic power series.

For instance, both f(x) + em1lz and f(x) have the same asymptotic power

series for x > 0.

We shall refer to the power series of Eq. (28) as the s t a n d a r d p r o t o t y p i c a l

a s y m p t o t i c p o w e r s e r i e s or, more simply, as the s t a n d a r d e x a m p l e of an

asymptotic power series.

3.4. Bard Sum of an Asymptotic Power Series

Even though only a finite few of the infinite terms of an asymptotic power

series are computationally useful for a partial sum, the unused terms con-

tain additional information about the function f(x). Borel26 discovered a

method that, when it works, uses all the terms in the series to compute

f(x) in principle exactly. At the same time it singles out precisely one of

the many functions associated with the same series in the Poincare sense.

Borel’s method is to obtain an integral representation from the series -

in the case of the example given above, to reverse the process that generated

5 9 8 M o d e r n E l e c t r o n i c S t r u c t u r e T h e o r y

t h e s e r i e s f r o m a n i n t e g r a l r e p r e s e n t a t i o n . T h e B o r e 1 s u m o f t h e p o w e r s e r i e s

c a n x n i s g i v e n l o o s e l y b y t h e f o r m u l a ,

F o r t h e s t a n d a r d e x a m p l e , a n = ( - I ) ? z ! , a n d t h e B o r e 1 s u m i s t h e o r i g i n a l

i n t e g r a l :

g ( x ) = l m e e t [ E ( - x t ) n ] c k t = i m e m t $ & d t . ( 3 4 )

A m o r e c a r e f u l d e f i n i t i o n o f t h e B o r e 1 s u m i s 2 7 :

( i ) S u p p o s e t h a t B ( t ) = x a n t n / n ! c o n v e r g e s i n s o m e c i r c l e 1 t i < 6 .

( i i ) S u p p o s e f u r t h e r t h a t - B ( t ) h a s a n a n a l y t i c c o n t i n u a t i o n t o a n e i g h b o r -

h o o d o f t h e p o s i t i v e r e a l a x i s .

( i i i ) S u p p o s e f i n a l l y t h a t g ( x ) = ( l / x ) J r e w t i z 1 3 ( t ) d t c o n v e r g e s f o r s o m e

x # 0 .

T h e n g ( x ) i s c a l l e d t h e B o r e 1 s u m o f t h e s e r i e s c a n x n , a n d B ( t ) i s c a l l e d

i t s B o r e 1 t r a n s f o r m .

N o t e v e r y a s y m p t o t i c p o w e r s e r i e s i s B o r e 1 s u m m a b l e . T h e m a i n t h e -

o r e m i s d u e t o W a t s o n , N e v a n l i n n a , a n d S ~ k a l . ~ ~ - ‘ ~ S u p p o s e t h a t t h e r e -

m a i n d e r e s t i m a t e

I N - l

h o l d s u n i f o r m l y i n N a n d z i n s i d e a c i r c l e C R o f r a d i u s R / 2 t a n g e n t t o t h e

y a x i s a t t h e o r i g i n , w i t h c e n t e r a t x = R / 2 ( t h a t i s , { z : R e z - l > E l } ) .

T h e n t h e B o r e 1 s u m o f t h e s e r i e s e x i s t s a n d e q u a l s f ( z ) f o r z i n s i d e t h e

c i r c l e C R . T h e r e a d e r i s r e f e r r e d t o t h e p r o o f i n t h e s h o r t p a p e r b y S o k a l .

T h e s t a n d a r d e x a m p l e c l e a r l y s a t i s f i e s ( 3 5 ) w i t h A = C T = 1 a n d i s B o r e 1

s u m m a b l e t o J a m e - ‘ ( 1 + x t ) - ‘ d t .

I t i s t h e e r r o r b o u n d ( 3 5 ) t h a t s i n g l e s o u t f ( z ) f r o m o t h e r f u n c t i o n s

t h a t h a v e t h e s a m e a s y m p t o t i c p o w e r s e r i e s i n t h e s e n s e o f P o i n c a r e . F o r

i n s t a n c e , i f j ( x ) s a t i s f i e s ( 3 5 ) , t h e n f ( x ) + e - l / % d o e s n o t , s i n c e i e - r i ’ [

a p p r o a c h e s 1 i f 2 a p p r o a c h e s 0 a l o n g a c u r v e t a n g e n t t o t h e i m a g i n a r y a x i s

Exact Expansion Methods for Atomic Hydrogen , . . 599

at the origin. That is, the error bound is violated for all N as x + 0 from

(almost) along the imaginary axis. If j(x) satisfies (35), then on the other

hand f(z) + e -‘/fi does not, because e --‘Ifi approaches 0 too slowly as

z -+ +0 along the real axis. That is, at x = l/(2N)2,

(N - l)! = $16NN!xN(N - l)!, (36)

and comparison with Eq. (35) shows that e-l/G has added a term that is

too large by a factor of (N - l)! to be accommodated by the error bound.

The Bore1 method, when it applies, connects a given power series with a

unique analytic function. Mathematical analysis carried out on the power

series is in this sense analysis carried out on the unique analytic function

associated with the series by Bore1 summation.

It is instructive to take the Bore1 sum of the geometric series with

alternating signs.

The integral clearly converges so long as Re (l/z) > -1; that is, everywhere

outside a circle of radius 1/2 centered at (x, 3) = (-1/2,0). The integral

can also be manipulated to converge everywhere except at ,z = -1. In such

a way, the Bore1 sum (or its analytic continuation) of the geometric series

can give the sum of the series at points outside the radius of convergence.

.5- Complex Bore1 Sum of a Real Asymptotic Power Series;

Stokes Line

The standard prototypical asymptotic power series (28) alternates in sign.

By changing the sign of x, one obtains a series in which all the terms are

positive:

1+ x + 2!x2 + 3!x3 + . . . . P%

600 Modern Electronic Structure Theory

Naive application of the Bore1 sum formula (33) yields

g( -x) = /- ,+ 0

cm 8 &dt .

The integrand has a singularity at t = l/x when x is real and positive, and

the formula is ambiguous. The limits obtained by taking x real from above

or below, however, are neither ambiguous nor identical:

1

e-t 1+ (-Lx f i&)t dt

1 -x - 1

= Pr e-t l-xt

-dt F ir% x *

Which of the two values is g( -x)? One must first decide whether to include

the negative real axis, which is a cut of the Borel-sum formula, with the

upper or lower half-plane. Once decided, one is led to the conclusion that

the sum of the series is complex, even though all the terms are real. The

sign of the imaginary part depends whether the real axis is approached

continuously from above or below. The negative real axis is a boundary of

the region on which the series (28) is Bore1 summable, and the boundary

can be treated only by continuity with the interior - but a choice must be

made of which interior.

Perhaps it is clearer to emphasize the analytic function to which the

standard example is Bore1 summable: x-iex-‘Ei(x-‘). The convergent

expansion ” for Er (z) has a log arithmic term,

00 (-,Z)n &(,z) = -T-lnZ---x------

nn! ’ (larg4 < 4. (41)

n=l

If the cut for In z is taken along the negative real axis, then

- ln 1

-x f i& EZO -1ni &i7r, (42)

X

Z x - l e - x - l C 1 F i 7 r x - 1 e - x - 1 .

n=l w

E x a c t E x p a n s i o n M e t h o d s f o r A t o m i c H y d r o g e n . . . 6 0 1

T h e a m b i g u i t y i n t h i s e x a m p l e a r i s e s f r o m a c u t i n t h e f u n c t i o n t h e a s y m p -

t o t i c e x p a n s i o n r e p r e s e n t s . I t m a y s e e m s t r a n g e t h a t a s e r i e s w i t h a l l t e r m s

r e a l l e a d s t o a f u n c t i o n t h a t i s c o m p l e x , b u t t h e s e r i e s i s b e i n g e v a l u a t e d o n

t h e b o u n d a r y o f i t s r e g i o n o f a p p l i c a b i l i t y , a n d b y a n a l y t i c c o n t i n u a t i o n .

T h e r e a d e r m i g h t o b j e c t a t t h i s p o i n t t h a t t h e b r a n c h c u t f o r t h e l o g a -

r i t h m n e e d n o t b e t a k e n o n t h e n e g a t i v e r e a l a x i s . I t c o u l d b e t a k e n a l o n g

a n y r a y f r o m 0 t o o o . T h e n J & ( z ) i s c o n t i n u o u s a c r o s s t h e n e g a t i v e r e a l

a x i s . W h a t h a p p e n s t o t h e e x p l a n a t i o n a b o v e - p a r t i c u l a r l y E q . ( 4 4 ) ?

l ? o r c o n c r e t e n e s s , l e t u s t a k e t h e l o g a r i t h m i c c u t j u s t b e l o w t h e p o s i t i v e

r e a l a x i s . T h a t i s , w e s p e c i f y & ( , z ) t o b e g i v e n b y E q . ( 4 1 ) , w i t h 0 5

a r g z < 2 7 r , s o t h a t , ? 3 r ( z ) i s c o n t i n u o u s a c r o s s t h e n e g a t i v e r e a l a x i s . T h i s

m e a n s a d i f f e r e n t b r a n c h o f . & ( z ) i n t h e l o w e r h a l f p l a n e t h a n b e f o r e . T h e

l o c a t i o n o f t h e c u t o f t h e f o r m u l a f o r t h e B o r e 1 s u m o f t h e s t a n d a r d s e r i e s

d o e s n o t c h a n g e : i t i s o n t h e n e g a t i v e r e a l a x i s . T h u s t h e s e r i e s r e p r e s e n t s

t h e

t h e

t h e

t h e

w r o n g b r a n c h o n ( 7 ~ < a r g z < 2 7 r ) . T o g e t t h e a s y m p t o t i c e x p a n s i o n f o r

r i g h t b r a n c h o f , 5 7 1 ( z ) , i t i s n e c e s s a r y t o a d d t h e d i s c o n t i n u i t y

b r a n c h e s . T h e c o r r e c t a s y m p t o t i c e x p a n s i o n f o r z e ’ _ ? $ ( z ) i s

s e n s e o f B o r e l ) ,

b e t w e e n

t h e n ( i n

z e ’ & ( z ) w g ( - z ) ? z ! , ( O < a r g z < r ) , n = O

, z e z l $ ( z ) - x(-z)+n! - 2 7 r i z e z , ( 7 r < a r g z < 2 n ) . W I

n = O

T h e a s y m p t o t i c e x p a n s i o n c h a n g e s f o r m d i s c o n t i n u o u s l y a c r o s s t h e n e g a t i v e

r e a l a x i s , b u t t h e f u n c t i o n i t r e p r e s e n t s i s c o n t i n u o u s . N e a r t h e n e g a t i v e

r e a l a x i s w h e r e 2 = - 1 ~ 1 , t h e a d d e d t e r m i s e x p o n e n t i a l l y s m a l l c o m p a r e d

w i t h t h e p r i m a r y s e r i e s . A d i s c o n t i n u o u s , e x p o n e n t i a l l y s m a l l c h a n g e i n t h e

a s y m p t o t i c e x p a n s i o n a t a l i n e w h e r e a l l t e r m s i n t h e s e r i e s a r e o f t h e s a m e

s i g n , a n d w h e r e t h e f u n c t i o n i t s e l f i s c o n t i n u o u s , i s t y p i c a l f o r a s y m p t o t i c

e x p a n s i o n s . S u c h a l i n e ( h e r e t h e n e g a t i v e r e a l a x i s ) i s c a l l e d a “ S t o k e s

l i n e ” o f t h e a s y m p t o t i c e x p a n s i o n . T h e B o r e 1 s u m ( w h e n a p p l i c a b l e ) g i v e s

a r a t h e r n i c e w a y t o u n d e r s t a n d S t o k e s l i n e s : t h e y a r e t h e c u t s o f t h e

B o r e l - s u m f o r m u l a w h e r e t h e a n a l y t i c f u n c t i o n i t s e l f h a s n o c u t .


. Dispersion Relation and Asymptotic ower-Series

Coeficients

The discontinuity across a single cut leads to a simple formula for the power-

series coefficients that is particularly useful for determining the asymptotics

of the RSPT energy coefficients in the LoSurdo-Stark effect and in the

anharmonic oscillator. Consider a function j(z) such that If(z) 1 + 0 as

1~1 --+ oo, that has a branch cut running from 0 to oo, and that is analytic in

the cut plane. One can use Cauchy’s formula, then deform the integration

contour to a large circle with a keyhole cut out for the branch cut. The

contribution on the large circle tends to zero as the radius tends to infinity,

and one obtains, if the branch cut is along the negative real axis,

1 Z-

J

00 f(-X -I- i0) - j(-X - i0) do

27ri 0 -x-z

1 -- = 27ri

m Af(+ dx x+z *

The discontinuity across the -x axis has been denoted by Aj(x):

Aj( -x) 3 j( -x -t i0) - j( -x - i0) . WI

An equation in the form of (50) is sometimes called a dispersion relation.

By expanding (x + z)-’ as a geometric series and integrating term by term

(without regard to the radius of convergence of the geometric series), one

obtains

J-(~) w 2 [-g lm x-N-lAf(-x)dx] zN . N=O

Exact Expansion Methods for Atomic Hydrogen . . 603

hat is, the power series coefficients are given by

(-l)N CG

aN = - 27ri s &v-Qf( --X)&X.

0

The function g(z) = .z --’ e ‘-‘&(zJ-~) that is the Bore1 sum of the stan-

dard example satisfies the two conditions on f(z). Its discontinuity is ob-

tained from Eq. (40),

-x -1

Ag(-XT) = -27riL. (54 X

Equation (53) then gives back the power-series coefficients exactly:

a* = (---I)* x-N-2e-zz-1 dx=(-l)*PJ!

That the dispersion-relation formula here gives the exact coefficient is a

consequence of having an exact, explicit expression for the discontinuity.

In the application below to the LoSurdwStark RSPT, the discontinuity is

only known asymptotically, and the dispersion relation yields an asymptotic

formula for the energy coefficients.

If the branch cut runs from 0 to +oo, then instead of Eqs. (50) and (53)

we would have

f(z) = $ /* Edx, (1m.z > 0) , 0

1 rzcl

aN = 2n-i 0 -J x -*-‘Af(x)dx.

4. Hydrogen Atom in Constant Electric

With the standard prototypical asymptotic expansion

to a physical problem to which it pertains.

oSurdo-Stark Eflect

in mind, we now turn

The electric analog of the Zeeman effect was observed first and published

second in 1913 by A. LoSurdo. ‘y3’ The second observation and first pub-

lication was by Stark. ’ In 1926, Schrcdinger ’ developed RSPT to give

a quantum mechanical explanation of the energy splitting, shifts, and

intensities.

ai4 Modern Electronic Structure Theory

4.2. RSPT Following SchrCdinger’s 1926 Treatment

This section owes much to Schrodinger’s original treatment.’ The

Schrodinger equation for atomic hydrogen in a constant external electric

field is

4.2.1. Parabolic Coordinates

Refore applying perturbation theory, Schrodinger took advantage of sepa-

rability and transformed to parabolic coordinates to “lighten our task.” In

more recent notation, ‘p14 the transformation is

z = (-2J?3)-1’2(O/#‘2 cos 4,

9 = (-2E)W112(crp)112 sin 4,

and the separated equations are

1 @l(C) = 0,

These equations are self-adjoint eigenvalue equations for the separation con-

stants ,Or and @J with volume elements cvldc and p-‘dp, respectively, and

are to be solved by RSPT. The perturbation parameter f is proportional

to F:

f = 3-2E)-3i2F. w

Exact Expansion Methods for Atomic Hydrogen . . 605

The energy is extracted from the separation constants via

After separation, Schrodinger applied RSPT to the eigenvalue equations for

the separation constants,

N=o

N=O

4.2.2. Algebruic Relations to Get E(F) from ,kll(f) and &(f)

Once the RSPT series have been obtained for ,& (f) and pz(j), some algebra

is required to get the RSPT series for E(F),

E(F) = 2 EcNkN, w N=O

because f depends implicitly on E via Eq. (65). It is helpful to deal with the

algebraic steps before the perturbative steps. First, f is found as a series

in F by series solution of the implicit equation that is the composition of

Eqs. (65) and (66),

Then successive substitutions into, and expansions of Eq. (66) yield

Eq. (69). Note that pi + p2 always appears as a sum. The formulae are

improved in appearance by introducing


We then find that

+ 6,+“~+p~~~~ + 7(“)2Y(3)]F3 + . . . . (76)

All this algebra takes place apart from the perturbation theory itself,

which is described in the next section. We give a few expbcit results here

for N 5 3. The parabolic quantum numbers nl and nz are non-negative

integers, and they are related to the magnetic quantum number m and

principal quantum number n via

n =w+w+m+l, vv

D[O) n-H-1 =nl+-

2 ’

tp mi-1

Z n2 i- -

2 ’


+ 9 ( m 2 - 1) n,+ (% y),

p = /g2) + & I = - 1 O n + 9n(m2 - 1 ) - 1 7 n 3 - 5 l n ( n l - n 2 ) 2 .

WI

T h u s o n e f i n d s t h a t

_Ecl) = in(nl - n2) ,

l d 2 J = kn4[-19 + 9m2 - 17n2 + 3(nl - n~a)~]. WJ

A l s o ,

_4?Zt3) = -&n7(nl - n2)[39 + llm2 + 23n2 - (nl - n2)2]. @f9

4 . 3 . C l o s e d F o r m T e r m - b y - T e r m S o l v a b i l i t y

o f t h e P e r t u r b a t i o n S e r i e s

T h e c o m p u t a t i o n o f t h e R S P T f o r t h e L o S u r d o - S t a r k e f f e c t f i n a l l y r e s t s o n

t h e R S P T f o r t h e p i e i g e n v a l u e e q u a t i o n s [ ( 6 3 ) a n d ( 6 4 ) ] , w h i c h d i f f e r o n l y

i n t h e s i g n o f f . O n e c a n w r i t e ,

d2 fI$)) E -r- +

m2 - 1

dr2 4 r

w h e r e t h e u p p e r s i g n i s f o r / 3 1 a n d t h e l o w e r f o r , & , a n d w h e r e t h e v o l u m e

e l e m e n t i s T-‘dT.

6 0 8 M o d e r n E l e c t r o n i c ~ i k u d u r e T h e o r y

F o r t h e r e a d e r w h o i s m o r e f a m i l i a r w i t h h a r m o n i c o s c i l l a t o r s t h a n h y -

d r o g e n a t o m s , t h e s e t h r e e e q u a t i o n s c a n b e p u t i n o s c i l l a t o r f o r m b y t h e

c o o r d i n a t e t r a n s f o r m a t i o n ,

T h e n $ ( r ) s a t i s f i e s

e m = _ k c - - z - L + m2 r2

2 d r 2 2r dr 2r2 - + y ,

E q u a t i o n ( 9 3 ) i s t h e r a d i a l H a m i l t o n i a n f o r a r a d i a l l y s y m m e t r i c , t w o -

d i m e n s i o n a l h a r m o n i c o s c i l l a t o r . E q u a t i o n ( 9 4 ) i s a q u a r t i c a n h a r m o n i c i t y .

T h u s t h e R S P T p r o b l e m f o r h y d r o g e n i n a n e l e c t r i c f i e l d i s e q u i v a l e n t t o

a n a n h a r m o n i c , r a d i a l l y s y m m e t r i c , t w o - d i m e n s i o n a l h a r m o n i c o s c i l l a t o r .

W h i c h e v e r p i c t u r e o n e p r e f e r s , t h e p e r t u r b a t i o n t i m e s a n y u n p e r t u r b e d

e n e r g y e i g e n f u n c t i o n g i v e s a l i n e a r c o m b i n a t i o n o f a t m o s t f i v e u n p e r t u r b e d

e i g e n f u n c t i o n s ( t w o u p a n d t w o d o w n ) , a n d t h e p e r t u r b e d w a v e f u n c t i o n

i n N t h o r d e r c a n i n v o l v e a t m o s t 4 N u n p e r t u r b e d e i g e n f u n c t i o n s . T h e

e n e r g y t u r n s o u t 4 t o b e a p o l y n o m i a l i n n , ( n l - n s ) , a n d m 2 w i t h r a t i o n a l

c o e f f i c i e n t s , a s i s i n f e r r e d f r o m E q s . ( 8 6 ) - ( 8 8 ) .

4 . 4 . R e c u r s i v e S o l u t i o n o f t h e P e r t u r b a t i o n S e r i e s t o H i g h O r d e r

T o c a l c u l a t e h i g h - o r d e r e n e r g y c o e f f i c i e n t s , c l o s e d f o r m p o l y n o m i a l s i n t h e

q u a n t u m n u m b e r s a r e n o t e f f i c i e n t . I t i s m o r e s t r a i g h t f o r w a r d t o i m p l e m e n t

t h e r e c u r s i v e R S P T e q u a t i o n s ( 8 ) a n d ( 9 ) f o r t h e s e p a r a t e d e q u a t i o n s e i t h e r

i n t h e h y d r o g e n i c ( E q s . ( 9 0 ) a n d ( 9 1 ) ) o r t w o - d i m e n s i o n a l a n h a r m o n i c o s -

c i l l a t o r ( E q s . ( 9 3 ) - ( 9 5 ) ) f a r m s . F o r i n s t a n c e , t h e n o r m a l i z e d e i g e n f u n c t i o n s

o f H i ” i n v o l v e g e n e r a l i z e d L a g u e r r e p o l y n o m i a l s t i m e s a n e x p o n e n t i a l :


Multiplication by r2 connects a zeroth order wave function with its four

closest neighbors:

With FORTRAN or Mathematics or Maple or the like, one can find the pi

RSPT series, and then the E( 8’) series.

4.5. Divergence;

Calculation of the

n = 1) yields4p3’

Complex Bad Sum

RSPT series for the ground state (nl = n2 = m = 0,

E(O) = -l/Z?, w

E(150J = -2.717977730 x lo28g, ow

E(N) N -(6/~)(3/2)~N!, (iv even), Pw

EtN) = 0, (iv odd) . w4

The numerical behavior as well as the asymptotic formula (101) show that

the series diverges. The factorial divergence suggests Bore1 summability.

Graffi and Grecchi5 proved that the RSPT series for the perturbed en-

ergy of any state for the LoSurdo-Stark effect is Bore1 summable for suf-

ficiently small IFI, so long as F is nut reul. This also constituted a con-

structive proof of the existence of resonance eigenvalues - i.e., complex

energy eigenvalues with eigenfunctions that are square-integrable in some

direction in the complex plane.

That the LoSurdo-Stark problem cannot have real eigenvalues can be

seen directly from a plot of the potential, ,zF - l/r, along the z axis. The

potential goes to -oo as z goes to -oo, and so the electron can always


tunnel through the barrier formed by the two terms and ionize. The eigen-

value spectrum changes from discrete (,!Z < 0) plus continuum (E > 0)

when there is no field (F = 0) to pure continuum when F # 0, no matter

how small _F is. The former bound states, however, do not disappear with-

out a trace. They turn into resonances. The energy eigenvalues move off

the real axis into the lower half of.the complex plane, while the correspond-

ing eigenfunctions are characterized by “purely outgoing wave” character.

The eigenfunctions are not square integrable so long as the coordinates are

real. But if the coordinates are rotated into the complex plane past cer-

tain threshold angles, the eigenfunctions are square integrable along the

appropriate rays. See Sec. 4.7 for details.

The complex resonance eigenvalues can be

Bore1 sum of the numerically obtained RSPT

cal detail is to find an analytic continuation of

calculated by taking the

series. The main practi-

the Bore1 transform valid

beyond the radius of convergence of its Bore1 associated series. One practi-

cal answer is to use Pad& approximants, by which a partial sum is replaced

by the quotient of two polynomials. A second, less delicate detail is to take

the integral in the Bore1 sum formula, in Eq. (33), along an appropriate

ray. By way of example, for the ground state with F = 0.10 a.u. (atomic

units), using RSPT through order 100, calculating the [25/25] Pade approx-

imants in F2, and carrying out the integration over t in Eq. (33) numer-

ically along the ray arg t = 7r/4, the value for the ground-state resonance

was found32 to be -0.527418176 - iO.007269056, compared to the exact

value, -0.52741817509. . . - iO.00726905676. . . .

4-6. Field-Ionization and Complex Energy Eigenvalues

The resonance eigenfunctions can play a role in approximate representations

of solutions of the time-dependent Schrodinger equation, at least not too

far from the origin, in regions of space for which the growth of the wave

function as the coordinate goes to oo has not yet set in. If the time-

dependent wave function is dominated by a single resonance eigenfunction,

then the probability density is given by


F o r t h e r e g i o n w h e r e t h i s f o r m u l a i s v a l i d , t h e p r o b a b i l i t y d e n s i t y d e c r e a s e s

e x p o n e n t i a l l y w i t h t i m e , w i t h t h e d e c a y c o n s t a n t ,

T h i s

p a r t

4 . 7 .

T h e

l / ~ ~ = - 2 I m & / l i . VW

i s t h e r e l a t i o n b e t w e e n t h e t u n n e l i n g i o n i z a t i o n r a t e a n d i m a g i n a r y

o f t h e c o m p l e x r e s o n a n c e e i g e n v a l u e .

V a r i a t i o n a l C a l c u l a t i o n o f C o m p l e x E n e r g y E i g e n v a l u e s

c o m p l e x e n e r g y e i g e n v a l u e s c a n b e a c c u r a t e l y c a l c u l a t e d v a r i a t i o n -

a l l y . ” F r o m E q . ( 6 3 ) , o n e c a n s e e t h a t f o r l a r g e 0 ,

a n d f r o m E q . ( 6 4 ) , t h a t f o r l a r g e p ,

I f t h e c o o r d i n a t e s a r e r o t a t e d s o t h a t o a n d p a r e r e p l a c e d b y o e i e l a n d p e i @ 2 r e s p e c t i v e l y , t h e n t h e o p t i m u m v a l u e s f o r 1 3 1 a n d 0 2 , f o r 8 ’ l a r g e e n o u g h

t h a t a r g f i s s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o , a r e

4 = - i a r g f ,

A p r o c e d u r e t h a t w o r k s w e l l i s t o f i n d t h e e i g e n v a l u e s o f t h e r o t a t e d v e r s i o n s

o f E q s . ( 6 3 ) a n d ( 6 4 ) ,

d 2 - e - i e l g - d u 2 + - ’

- i e l m 2 - 1

40 +e

- i e 2 d 2 -e -

‘ d p 2 + - ’ - i & m 2 - 1 + e i t 1 2 P - -

4 p 4 e 2 i e 2 f p 2 - b 2 @ z ( p ) = 0 , ( 1 1 1 )

i n a b a s i s o f t h e u n p e r t u r b e d f u n c t i o n s . A n i m p o r t a n t d e t a i l i s t h a t i n

t h e s c a l a r p r o d u c t s , t h e l e f t - h a n d f u n c t i o n i s n o t c o m p l e x c o n j u g a t e d . T h e

612 M o d e r n E l e c t r o n i c S t r u c t u r e T h e o r y

m a t r i c e s a r e n o t s e l f - a d j o i n t , a n d e i g e n v a l u e r o u t i n e s t h a t c a n d e a l w i t h

g e n e r a l c o m p l e x m a t r i c e s a r e r e q u i r e d . T h e p r o c e d u r e i s s i m p l e e n o u g h t o

d o o n a d e s k t o p c o m p u t e r . S e e R e f . 2 2 f o r d e t a i l s .

A s a n e x a m p l e , l i s t e d i n T a b l e 1 a r e r e s o n a n c e e i g e n v a l u e s f o r a l l 1 9

( n = 1 9 , m = 0 ) r e s o n a n c e s i n a f i e l d o f 5 7 1 4 V / c m t h a t w a s p a r t o f

a t h e o r e t i c a l s t u d y 2 2 1 3 3 o f a p h o t o i o n i z a t i o n e x p e r i m e n t . r 7 N o t e t h a t t h e

h i g h e s t e n e r g y r e s o n a n c e , (nl , n 2 , m ) = ( 1 8 , 0 , 0 ) , h a s a n i m a g i n a r y p a r t t o o

s m a l l t o b e o b t a i n e d i n t h e v a r i a t i o n a l c a l c u l a t i o n . T h e l i m i t i n g f a c t o r i n

t h e v a r i a t i o n a l c a l c u l a t i o n i s t h e n u m e r i c a l p r e c i s i o n o f t h e e n t i r e c o m p l e x

n u m b e r , a n d f o r t h a t s t a t e t h e r e w a s a 1 0 1 6 r a t i o i n m a g n i t u d e s b e t w e e n

t h e r e a l a n d i m a g i n a r y p a r t s . F o r t h e l i m i t P - + 0 , a n o t h e r a p p r o a c h i s

n e e d e d t o c a l c u l a t e t h e i m a g i n a r y p a r t s o f t h e r e s o n a n c e e n e r g i e s .

Table la Data for (n = 19, m = 0) resonances at F = 5714 V/cm (1.1123 x 10d6 hydro- genie atomic units): resonance eigenvalues in cm-l, and lifetimes in p.s. (1 hydrogenic atomic unit of energy = 2 x 109677.583 cm-l. See Ref. 22.)

k = (mmm) Re Ek (cm-l) -1m E b (cm-l) lifetime (ps)

(C41&0) -450.91 3.10 8.56 x lO-7

(1,177 0) -436.58 2.36 1.13 x 10-6

(2,16> 0) -422.13 1.71 1.56 x lO-6

(3,15,0) -407.58 1.16 2.30 x lo-‘j

(4,14,0) -392.96 0.714 3.72 x lo--c’

(5,133 0) -378.28 0.391 6.80 x 10-6

(6,12,0) -363.58 0.181 1.46 x lo-r’

(7,llYO) -348.90 6.87 x lO-2 3.86 x lO-5

(8,16,0) -334.24 2.07 x lO-2 1.28 x lo-*

(999,O) -319.61 4.94 x 10-3 5.38 x lO-4

(10,8,0) -304.99 9.27 x 10-4 2.86 x lO-3

(11,7>0) -290.38 1.36 x lO-4 1.95 x 10-2

(12,6,0) -275.76 1.52 x lO-5 0.175

(13,5,0) -261.12 1.25 x lO+ 2.12

(14>4,0) -246.47 7.24 x lO-8 36.7

(15>3> 0) -231.80 2.69 x lO-g 986.

(16,2,0) -217.11 5.69 x lo-l1 4.66 x 104

(17, LO) -202.40 5.40 x 10-13 4.91 x 10s

(18,0,0) -187.68 < 1.00 x 10-14 > 2.00 x 10s

Exact Expansion Methods for Atornic Hydrogen . . . 613

4.8. Perturbation Theory for the Imaginary Tewns

In this section we obtain a small-F expansion for the imaginary part of the

resonance eigenvalues.

In the same way that g( --x+i&) -g( --z--G) # 0 and is purely imaginary

and exponentially small in Eq. (40), so AE = ,?!Z(F + i&) - E(F - i&) # 0

and is purely imaginary and exponentially small on the real axis in the

LoSurdo-Stark effect. One-half the discontinuity is the imaginary part of

the resonance eigenvalue E. The discontinuity in the energy at real F

arises from the discontinuity in the separation constant pz at real, positive

f . It is the ,& equation (64) that has the tunneling barrier, not the pi

equation (63).

The small-f expansion for the discontinuity in &, (denote it A@J) turns

out to have the form15

w2 N -+2!(?22 + [T7L~)!]-1_f- 2~rb’+~~-~/@f~(~ + b(l)j + /&292 +. . . ).

That is, it is f to a negative power times an exponentially small factor

times a power series in f . To get AE into a similar form, but with respect

to F, it is necessary to unravel Eqs. (65)-(68) and (112). We treat the

unraveling first, after which the calculation of A/?2 will be discussed.

Consider for a moment Eqs. (45) and (46) for zez& (z). These are the

asymptotic expansions for a branch that is continuous across the negative

z axis. The form of the asymptotic expansion changes discontinuously,

because the (Bore1 sum of the) power series refers to different branches.

The exponential small subseries (a single term in Eq. (46)) represents the

discontinuity between the branch that is continuous from above with the

branch represented by the Bore1 sum below.

There is an analogous situation for b2: that is, for Im f > 0 the

power series (68) Borel-sums to one branch, and for Im f < 0 to another.,

If the analytic continuation is taken from above to below, then an

additional exponentially small subseries AD2 gets added to the RSPT

series, and the exponentially small subseries is a direct expansion for

the discontinuity.

An exponentially small subseries in @J leads to an exponentially small

subseries in E through Eq. (66), and further an exponentially small sub-

series in f(F) through Eq. (65). The unraveling requires keeping terms


that are zeroth and first or&r in exponentially small factors. To begin,

2(pr(f+ Af) + &+ Af) + A&)2

1

-2@r + p2)2 *

(113)

What is meant by an over-bar is the RSPT expansions (67) and (68) for ,&

and ,&, (74) for f, and (69) for J!?. The exponentially small contribution to

Af is

Af c A ;(-2E)-3i2F 3AE-

= -5xf = 3(& + Dz)~~AE. (114)

The discontinuity AE in E is then given by

432

N @I + $2)3 - f(cz/dj)@~ + P2)3 + oKAp2)21 * ( 115)

The problem of finding A E( 8’) consists in first finding A&(j), next finding

AE(f) from, Eq. (115), and finally using Eq. (74) (reversion of Eq. (70))

for j(F) to get AE as a function of F.

To calculate AD2 is the crux of the problem. One can exploit that AD2

is purely imaginary for small real f, and obtain a current-density formula

from the eigenvalue equation (64):

The trick is to use this formula for p inside the classically forbidden bar-

rier region that extends from (ignoring m) approximately p = 4/?z to

p = 1/(4f). The numerator is what picks out the exponentially small imag-

inary contribution from the larger real contribution. The resonance bound-

ary condition, that @z have purely outgoing wave character (Eq. (107)), is

crucial to the numerator. The denominator, however, gets its main contri-

bution from the non-small, essentially real part of the wave function.


T o c a l c u l a t e & a n d i m p o s e t h e b o u n d a r y c o n d i t i o n a t o o , o n e n o t e s

t h e a n a l o g y b e t w e e n E q . ( 6 4 ) a t l a r g e p a n d t h e l i n e a r - p o t e n t i a l p r o b l e m

w h i c h s u g g e s t s a J W K B - l i k e m e t h o d u s i n g f i n s t e a d o f h a s t h e e x p a n s i o n

p a r a m e t e r . A c c o r d i n g l y , w e f i r s t t r a n s f o r m t h e e i g e n v a l u e e q u a t i o n t o m a k e

t h e o u t e r t u r n i n g p o i n t i n d e p e n d e n t o f f ,

d 2 - 6 4 f 2 G + 1 6 f 2 m ; ; ’

P 2 - 1 6 j - + l - x Q 2 = 0 .

I W J

X

T h e o u t g o i n g - w a v e J W K B - l i k e s o l u t i o n i n t h e c l a s s i c a l l y a l l o w e d i o n i z a t i o n

r e g i o n h a s t h e f o r m ,

T h e r e a l a x i s t o t h e l e f t o f t h e t u r n i n g p o i n t i s a S t o k e s l i n e o f t h e J W K B

e x p a n s i o n , a n d t h e f o r m o f t h e w a v e f u n c t i o n d e p e n d s o n w h e t h e r I m x

i s > 0 o r < 0 ( R e f . 3 4 ) . T h e J W K B “ c o n n e c t i o n - f o r m u l a p r o b l e m ” i s

d i s c u s s e d i n S e c . 6 . 4 . N o t e t h a t & , h e r e c o r r e s p o n d s t o Q i n S e c . 6 , w h i l e

S a c o r r e s p o n d s t o S . T h e S b i s u s e d h e r e i n s t e a d o f Q t o k e e p t h e n o t a t i o n

a s c l o s e a s p o s s i b l e t o t h a t o f R e f . 1 5 . T h e r e s u l t s t h a t w e n e e d a r e g i v e n

i n E q s . ( 2 1 6 ) a n d ( 2 1 7 ) w i t h a = i b = i / a . T h e q u a n t i t y 1 - x h e r e

c o r r e s p o n d s t o - x i n E q . ( 2 1 6 ) - ( 2 1 8 ) . W h e n x i s i n t h e c l a s s i c a l l y f o r b i d d e n

r e g i o n a b o v e t h e r e a l a x i s ,

(14 +z 1 , a r g ( 1 - x ) N 0, Imx > 0, I m ( 1 - x ) < 0 ) ( 1 2 1 )

( t h e r e d u n d a n c y o f t h e s t a t e d c o n d i t i o n s i s t o e a s e c o m p a r i s o n w i t h

E q . ( 2 1 7 ) ) , a n d b e l o w t h e r e a l a x i s ,

+4 +c 1 , a r g ( l - x ) w O , I m x < O ) . ( 1 2 2 )


The Sa and Sb satisfy the Riccati equations,

P2 =x-1+16f;

-64f2 [m;;’ - (9+1’2 -& ($$‘)-1’2] , (123)

+64f2 [ml;’ - (-$9+1’2 & (-9-1’2] . (124)

For large x, Sa N $(x - l)3j2. For srndl x, Sb - $!(l - x)~/~. In fact,

If the numerator of Eq. (116) is evaluated to the right of x = 1, then

Eq. (120) yields to zeroth exponential order a constant:

The same result can be obtained by evaluating the numerator to the

left of x = 1, but the calculation is more subtle, because the real axis is

a stokes line. It is necessary to stay off the real axis. For instance, first

take Im x > 0, which selects the expression given in Eq. (121). Complex

conjugation, however, takes Im x > 0 into Imx < 0 for the conjugated

expression and switches branches (compared with Eq. (122)).

P 2,m, (1x1 a 1, arg(1 -x) - 0, hx = +O)]*

N fi ( 2) -“’ esb/t’-fj, (1x1 < 1, arg(1 - x) N 0, hx = -0).

(Put another way, the right-hand side of Eq. (127) is the expression for

the incoming wave solution, just below the real axis in the barrier region).


To calculate the numerator, it is necessary to use the same “x” in all the

expansions: that is, to use Eq. (122) for @z,- and Eq. (127) for @&,, both

corresponding to Im x = -0. The result is the same as (126).

To calculate the denominator in Eq. (116), it is necessary to obtain

explicit expressions for ,S’b. We expand ,!$, in a power series in f:

S&r, f) = 2 &,iv(x).P * Gw

N=O

(Again, the notation is being held close to that to Ref. 15 from which this

calculation is taken. In Sec. 6, Sb,N is denoted QcN).) Series solutions of

Eq. (124) then yields

&,-J(x) = $1 - x)3/2 ,

$,2(x) = (1 - x)“~ [ ( -32&sJ2 + 8(m2 - 1)) x-r

+64&j2(l -x)-l ” -- 3 (1 -X

2 )- ] ,

and so forth. Recall that the upper limit p in the integral in the denominator

of Eq. (116) is inside the barrier so that p < 1/(4f), which is the same as

x < 1. For small x,

&o(x) = ; -x+-x2+... , ;

&l(x) = 8/3$O’ ln ;+8&’ ;x+;x2+... C

,

&#,2(X) = -32j3ioJ2 + 8(nz2 - 1)

+ C

T + 80&‘2 - 4m2 X

+ 11 + 36@$“‘2 - m2


Next we put these expressions for &,(j,~) into Eq. (121), substitute x =

4fp, and obtain a small-f expansion. If one looks just at &,u + fS’b,i in

esbic8f), one finds that

This has the form of &~~?‘e1~(i2~)~7z2!(~2 + [ml)! times the highest

power term (#r~“‘eW~~2) in the unp erturbed wave function. One can show

in fact that

where N(f) is a relative normalization factor, and @z,nspT is the RSPT

wave function.

Putting together Eqs. (116) for A&?, (126) for the numerator, and (136)

for the denominator, one obtains

In the term-by-term evaluation of the integral in the denominator, it is

permissible to put oo for the upper limit (the error is exponentially small),

and it is also necessary to evaluate the relative normalization factor N(f).

The details (cf. Ref. 35) can be found in Ref. 15. The form of the result is

The form of the discontinuity in E follows from Eqs. (138), ( 115) for

AE in terms of A&, and (74) for f(F):

-2nz-b~-1 e-2/(3n3)F+3(nl -nz)

AE n3n2!(n2 + l7nl)!

(1+u %‘+d2)F2 + . . . ) .

( 139)

4.9. Dispersion Relation and Asymptotics of RSPT

Discontinuities across a cut suggest dispersion relations. The energy E(F)

does not vanish as I F[ -+ co, but Fe2 E does, which is sufficient to obtain

the final formulae for the expansion coefficients. Similarly for ,Gi and &.


Consider first ,&(f). The discontinuity (Eq. (138)) is across the positive

f axis, so that Eqs. (56) and (57) are relevant, and one obtains

f-jN+2n2+~m~+1 (N + t&z2 + jm[)! 1 + &?Jrn /6 N--

2lr ns!(n2 + lml)! N + 2n2 + \mi

bpim /62

‘(N+2n2+jml)(N+2n2+[m[--l)‘*** * 1 The roles of ,Br and /32 are interchanged by changing the sign of f. Conse-

quently the asymptotics of the RSPT @iNJ are given by

fjN+2nl+imi+1 (N + 2nl + [ml)! /p m(-ly+l 2r

1 + b?)rn/G

nl!(nl + lml)! N + 2nl+ lml

The energy B(F) has cuts both at positive F and at negative F. The

discontinuity at negative F can be obtained from the discontinuity at posi-

tive F (Eq. (139)) by recalling that changing the sign of F interchanges the

two quantum numbers nl and n2. The dispersion-relation based formula

for the RSPT energy coefficients accordingly has two contributions:

E(N) ru nlnlm

@n2+jmi+l 3 c I Tin

3 N e3(nr-n2)

-

27rn3n2!(n2 + /ml)! (N + 2n2 + lrnl)!

x d7!Jn2m/( $n3) a?)n2m/( in3)2

’ -t N + 2n2 + lrnj -k (N + 2n2 + lml)(N + 2n2 + lrn[ - 1) -t ’ * *

-(-l)N fjW+lmj+l 3

c 1 P 3 N e3(n2-nl)

27rn3nl!(nl + lrnl)! (N + 2nl+ ]mj)!

620 Modem ElectTonic &‘mm3we Theory

From Eq. (143), one sees that the growth of the RSPT coefficients is fac-

torially fast, with the dominant behavior depending on whether ni or n 2

is larger ((N + 2n> + jml)!). S ome values of the coefficients uml nZ m are Ck)

given in Table 2. A comparison of EiyiZm calculated by RSPT versus the

asymptotic formula (143) is given in Table 3.

5. P h o t o i o n i z a t i o n o f A t o m i c H y d r o g e n

The RSPT and JWKB expansions discussed in earlier sections involved

power series. In this section we touch briefly on the expansion of the

photoionization cross-section in terms of resonance contributions. This

expansion is not a power series expansion, but a Mittag-Leffler expansion:

a sum over the poles of a function.

T a b l e 2 . V a l u e s o f t h e c o e f f i c i e n t s c & “ ‘ ~ ~ ~ t h a t o c c u r i n t h e a s y m p t o t i c f o r m u l a ( 1 4 3 )

f o r d & & m , t a k e n f r o m R e f . 1 5 .

n l w m k $4 nlnzm

0 0 0 0 1

1 - 8 . 9 1 6 6 6 6 6 6 6 6 6 6 6 7

2 2 5 . 5 6 5 9 7 2 2 2 2 2 2 2 2

3 - 1 5 8 . 7 4 6 4 3 1 3 2 7 1 6 0

4 4 6 . 9 0 4 8 6 8 5 4 5 8 4 6 2

5 - 1 0 2 7 3 . 1 7 2 8 3 3 4 4 4 5

1 0 0 0 1

1 - 1 7 3 . 3 3 3 3 3 3 3 3 3 3 3 3

2 1 4 6 9 0 . 2 2 2 2 2 2 2 2 2 2

3 - 1 1 4 6 4 9 2 . 8 3 9 5 0 6 1 7

4 9 2 3 9 4 9 9 9 . 0 4 5 2 6 7 5

5 - 8 9 3 2 6 3 6 9 3 3 . 9 3 4 7 1

0 1 0 0 1

1 - 1 8 9 . 3 3 3 3 3 3 3 3 3 3 3 3

2 1 2 7 3 5 . 5 5 5 5 5 5 5 5 5 5

3 - 6 5 9 8 0 8 . 3 9 5 0 6 1 7 2 8

4 3 0 2 1 4 3 3 5 . 1 4 4 0 3 2 9

5 - 2 9 9 2 6 9 3 9 0 2 . 3 6 2 6 9

Table 3. Comparison of Eiy,&n calculated by Rayleigh-Schrtidinger perturbation theory and by the asymptotic formula

(Eq. (143)) with the use of the r_~~~‘~~~ given in Table 2.

EL:&* by asymptotic expansion to term k Ek:Azm by

Order k=O kc3 k=5 RSPT

ground state: nl = 0, nz = 0, m = 0

10 -3.996 474 085 x108 -1.864 273 316 x108 -1.692 830 016 x108 -1.945 319 605 x108

30 -9.714 025 167 ~10~~ -7.897 322 796 ~10~~ -7.896 691 154 ~10~~ -7.897 811 108 ~10~~

50 -3.703 729 008 ~10~~ -3.279 092 637 ~10~~ -3.279 079 139 xlo73 -3.279 134 470 xlo73

70 -4.850 589 779 ~10"~ -4.449 391 053 x10112 -4.449 388 577 ~10"~ -4.449 406 910 ~10"~

90 -2.000 554 940 xlOl54 -1.871 123 931 ~10'~~ -1.871 123 720 ~10'~~ -1.871 126 438 ~10'~~

110 -7.111 439 484 ~10"~ -6.733 615 461 ~10"~ -6.733 615 281 ~10"~ -6.733 619 559 ~10"~

130 -9.628 490 607 ~10~~~ -9.194 526 355 ~10~~~ -9.194 526 303 ~10~~~ -9.194 529 248 ~10~~~

150 -2.828 678 874 ~10~~' -2.717 977 245 ~10~~' -2.717 977 245 ~10~~' -2.717 977 730 ~10~~'

excited state: nl = 0, n2 = 0, m = 0

10 -6.883 958 372 ~10'~ -2.931 152 857 ~10'~ -1.051 598 712 ~10'~ -1.247 119 323 ~10'~

25 2.186 883 327 ~10~~ 1.136 023 636 ~10~~ 1.140 631 715 ~10~~ 1.139 904 838 ~10~~

40 -4.446 779 891 x10g3 -2.980 690 505 x10g3 -2.982 611 712 x10g3 -2.982 601 906 x10g3

55 1.956 524 041 ~10'~~ 1.464 777 098 ~10'~~ 1.465 028 714 ~10'~~ 1.465 028 976 ~10'~~

70 -4.580 728 565 ~10'~~ -3.651 445 885 ~10'~~ -3.651 684 658 ~10'~~ -3.651 685 117 ~10'~~

85 2.420 337 907 ~10~~~ 2.008 508 544 ~10~~~ 2.008 567 784 ~10~~~ 2.008 567 875 ~10~~~

100 -1.705 384 915 ~10~~' -1.455 619 001 ~10~~' -1.455 641 369 ~10~~' -1.455 641 398 ~10~~'

115 1.082 850 832 x103= 9.436 075 763 x10315 9.436 157 808 x10315 9.436 157 893 x10315

130 -4.706 327 872 ~10~~~ -4.167 086 679 ~10~~~ -4.167 108 834 ~10~~~ -4.167 108 853 xlO363

145 1.118 686 686 xl0412 1.003 083 772 x10412 1.003 087 195 x10412 1.003 087 198 x10412

6 2 2 M o d e r n E l e c t r o n i c , ! h - u c t w e T h e o r y

. l . C o m p l e x E n e q i e s A n ? N o t E n o u g h

T h e p h o t o i o n i z a t i o n c r o s s - s e c t i o n i s m e a s u r e d b y t a k i n g a b e a m o f h y d r o -

g e n a t o m s p r e p a r e d i n a p a r t i c u l a r i n i t i a l s t a t e , i r r a d i a t i n g t h e b e a m w i t h

a p o l a r i z e d t u n a b l e l a s e r , a n d t h e n c o u n t i n g t h e n u m b e r o f e l e c t r o n s o r

p r o t o n s p r o d u c e d b y i o n i z a t i o n . A l t h o u g h t h e l o c a t i o n o f s h a r p f e a t u r e s

a n d t h e i r w i d t h s ( w h e n b r o a d e r t h a n t h e i n h e r e n t i n s t r u m e n t a l w i d t h ) c a n

b e o b t a i n e d f r o m t h e r e a l a n d i m a g i n a r y p a r t s o f t h e c o r r e s p o n d i n g r e s -

o n a n c e e i g e n v a l u e s - e i t h e r b y R S P T , v a r i a t i o n a l c a l c u l a t i o n , o r o t h e r

m e t h o d s - n e i t h e r t h e a s y m m e t r y o f t h e s p e c t r a , t h e r e l a t i v e i n t e n s i t i e s

o f t h e f e a t u r e s , n o r t h e c o n t r i b u t i o n s o f b r o a d r e s o n a n c e s c a n b e o b t a i n e d

f r o m t h e r e s o n a n c e e i g e n v a l u e s a l o n e .

5 . 2 . E x p a n s i o n a s S u m o v e r C o m p l e x R e s o n a n c e s :

l h z n s i t i o n D i p o l e s

T h e b a s i c f o r m u l a f o r t h e p h o t o i o n i z a t i o n o f a n i n i t i a l s t a t e $ 0 t o a c o n -

t i n u u m s t a t e @ E i s g i v e n i n r e d u c e d - m a s s a t o m i c u n i t s b y a g o l d e n - r u l e

e x p r e s s i o n ,

H e r e c i s t h e v e l o c i t y o f l i g h t , a n d r . i S d e n o t e s t h e c o o r d i n a t e i n t h e

d i r e c t i o n o f t h e p o l a r i z a t i o n o f t h e p h o t o n o f f r e q u e n c y ~ / 2 7 r . T h e e n e r g y

E a n d t h e p h o t o n f r e q u e n c y a r e c o n n e c t e d b y t h e B o h r f r e q u e n c y r e l a t i o n ,

E - E o = h i . I n h y d r o g e n , t h e c o n t i n u u m s t a t e s a r e a l s o i n d e x e d b y t h e

p a r a b o l i c q u a n t u m n u m b e r n l a n d t h e m a g n e t i c q u a n t u m n u m b e r m , w i t h

e a c h c o n t i n u u m m a k i n g a c o n t r i b u t i o n t o t h e p h o t o i o n i z a t i o n c r o s s - s e c t i o n :

A n u m e r i c a l d i f f i c u l t y i n u s i n g t h e g o l d e n r u l e f o r m u l a i s t h e e x q u i s i t e

s e n s i t i v i t y t o E i n t h e n e i g h b o r h o o d o f a s h a r p r e s o n a n c e .

T h e p h o t o i o n i z a t i o n c r o s s - s e c t i o n c a n b e e x p r e s s e d a s a s u m o f r e s o -

n a n c e c o n t r i b u t i o n s . T h e m e t h o d i s s i m i l a r t o t h e d e r i v a t i o n o f t h e d i s -

p e r s i o n r e l a t i o n i n S e c . 3 . 6 : e x t e n d t h e g o l d e n r u l e e x p r e s s i o n f o r o ( E ) t o

o n e t h a t i s a n a l y t i c ( i . e . , e l i m i n a t e c o m p l e x c o n j u g a t i o n ) ; a p p l y C a u c h y ’ s

i n t e g r a l f o r m u l a ; e x p a n d t h e c o n t o u r t o a l a r g e c i r c l e . I n t h i s c a s e , h o w e v e r ,

Exact Expansion Methoda for Atomic Hydrogen . . . 623

there is no branch cut - only poles. The contributions from the upper half

plane are the negative of the complex conjugates of those from the lower,

which in turn are the sum from the residues at the poles. Each pole is

a resonance eigenvalue &1n2m which can be indexed by its unperturbed

parabolic quantum numbers, nr, n2, m. In such a way, Eq. (145) becomes22

Here B(E) is a possible background term (coming from integration over

the large circle whose radius tends to CQ), and ,x& n2m is the square of a

complex transition dipole matrix element,

The use of parentheses rather than angular brackets is to indicate that no

complex conjugation is to be taken. The use of “-m” replaces complex

conjugation for the C$ variable. The integration has to be taken not on the

real CY and p axes, but along rays in the complex plane, as indicated by the

angles 191 and t9z in Sec. 4.7. The resonance

according to

(@ nln2,-mlhln2mJ

eigenfunctions are normalized

= 1. (148)

A most important feature of the contribution of each resonance is that the

transition-dipole-squared can be complex, Consequently, the contribution

is inherently asymmetric:

1m ptnln2m Irn Enln2mRe PG,nln2m

E - Enln2rn = (E - ReEnln2mJ2 + (Im&ln2m)2

The first term is the usual symmetric, absorptive Lorentzian. The second

term is nonzero if the transition dipole is significantly complex, and it has

the shape of an antisymmetric, dispersive Lorentzian.


5 . 3 . C o m p a r i s o n w i t h E x p e r i m e n t

W e s h o w h e r e c o m p a r i s o n s w i t h t w o e x p e r i m e n t s .

O n e e x p e r i m e n t l 6 i n v o l v e s t r a n s i t i o n f r o m t h e g r o u n d s t a t e t o a s i n g l e ,

i s o l a t e d ( n = 4 ) f i n a l s t a t e i n a v e r y l a r g e e l e c t r i c f i e l d o f 2 . 6 1 M V / c m .

E s p e c i a l l y i n t e r e s t i n g i n t h e e x p e r i m e n t i s t h a t t h e h i g h - e n e r g y w i n g d o e s

n o t r e t u r n t o t h e s a m e b a s e l i n e a s t h e l o w - e n e r g y w i n g - t h a t i s , t h e l i n e

i s a s y m m e t r i c , T h e a s y m m e t r y , i n t h e l i g h t o f E q . ( 1 4 9 ) , i n d i c a t e s t h a t t h e

e x c i t e d , i o n i z i n g r e s o n a n c e i s a p p r e c i a b l y c o m p l e x , a n d c o n s e q u e n t l y n o t

c o n v e n i e n t l y c a l c u l a t e d b y R S P T . A v a r i a t i o n a l c a l c u l a t i o n o f t h e ( 0 3 0 )

r e s o n a n c e , E q . ( 1 4 9 ) , p l u s c o n v o l u t i o n w i t h a n e x p e r i m e n t a l l y d e t e r m i n e d

i n s t r u m e n t a l l i n e - s h a p e f u n c t i o n , r e p r o d u c e d t h e e x p e r i m e n t a l c u r v e , a s

s h o w n i n F i g . 1 .

1 2 . 5 2 1 2 . 5 0 1 2 . 4 8 1 2 . 4 6 1 2 . 4 4 1 2 . 4 2 1 2 . 4 0 1 2 . 3 8

e x c i t a t i o n energy ( e V )

F i g . 1 . C o n t r i b u t i o n o f t h e ( 0 , 3 , 0 ) r e s o n a n c e t o t h e p h o t o i o n i z a t i o n c r o s s - s e c t i o n o f t h e g r o u n d s t a t e o f h y d r o g e n i n a f i e l d o f 2 . 6 1 M V / c m : + w i t h e r r o r b a r s i n d i c a t e e x p e r i m e n - t a l p o i n t s f r o m B e r g e m a n e t & 1 6 ; t h e n a r r o w e r s o l i d c u r v e i s c a l c u l a t e d v i a E q . ( 1 4 6 ) f r o m t h e u b i n i t i o r e s o n a n c e e i g e n f u n c t i o n ; t h e b r o a d e r s o l i d c u r v e i s a f t e r c o n v o l u - t i o n w i t h t h e i n s t r u m e n t a l l i n e - s h a p e f u n c t i o n . T h e v e r t i c a l s c a l e a n d b a s e l i n e o f t h e c o n v o l u t e d p l o t w e r e d e t e r m i n e d b y l e a s t - s q u a r e f i t t i n g t o t h e e x p e r i m e n t a l p o i n t s . T h e v e r t i c a l s c a l e a n d b a s e l i n e o f t h e n o n - c o n v o l u t e d p l o t a r e c o n s i s t e n t w i t h t h e c o n v o l u t e d .


2 ZJ L=i

l I

2.5

2

1.5

2 I t

0.5

0

-0.5

0

.

, , 100 0 - 100 - 200 - 300 -400

ENERGY (cm-’ )

I I I I

I 1 1 I 1 1

100 0 -100 -200 -300 -400

Energy (any')

Fig. 2. (a) Experimental photoionization spectrum at F = 5714 V/cm with r-polariza- tion from the n = 2 state that has high-field parabolic quantum numbers (l,O, 0), from Rottke and Welge.17 (b) Theoretical ab initio photoionization spectrum that is purely the superposition of resonance contributions.


The second experiment17 involves excitation of hydrogen in an (n = 2)

state to states with n 2 18, all in an electric field of 5714 V/cm. The

spectrum is quite complicated, with sharp lines, slightly broad asymmetric

lines, with “apparent” background, and with “field-induced modulations”

above the field-free ionization energy. As shown in Fig. 2, all this can

be reproduced by the addition of individual resonance contributions -

the broadest of which contribute to what otherwise looks like continuum

background.

6. JWKB Connection Formulae at a Linear Turning Point

The usefulness of the JWKB method was demonstrated earlier in Sec. 4.8.

The main formal problem is not the calculation of the wave function, but

the connection of the formulae for the wave functions that are asymptotic

in nature and that are valid in different regions. The derivation of the

connection formulae at a linear turning point is treated in most elementary

quantum textbooks. Most do not recognize the Stokes-line nature of the z

axis inside a classically forbidden region, and the formulae that result are

often ambiguous and often described as being uni-directional rather than

bi-directional. The approach taken here34 goes through the Airy function

asymptotic expansions, which are shown to be Borel-summable. conse-

quently the JWKB expansions are in one-to-one correspondence with the

wave functions they represent, and there is no restriction as to directionality

of the connection formulae.

6.1. JWKB Wave Function

The JWKB method puts the wave function in the form

-112 eLs(z,h)/h

7 whereI3--V>O, (150)

$ m (-(*)~)-1’2e~Q(~~~)/~, where E-V < 0. (151)

The (3~) inside the square roots in Eqs. (150) and (151) is to be taken to

make Q and S positive at the turning point: (+) if the classically allowed

region is to the right, (-) if to the left. Then S and Q are expanded as


power series in h2.

S(x, fi) = S(O)(x) + IW1)(z) + . . . , (152)

Q(x, Ii) = Q(‘)(x) + li2Q@)(x) -/- . . . . (1W

The equations satisfied by S and Q are then, with the (+) sign taken for

convenience,

dS@) - = ; [ 2 4 E - v ) ] - 1 ~ 4 $ J 2 7 n ( E - V ) ] - 1 ' 4 , dx

UJw

dQ(l) = 324~ - E)]-1~4-$2m(v - E)]-1’4. dx

The representation of the wave function is invalid near a classical turn-

ing point, where V = E , because of singularities in S and Q. For concrete-

ness, take E - V > 0 to the right of the turning point:

Then,

2 m ( E - V ) = q ( x - Xl) + q(x - m)21, q>o. w-w

dSc”)

dx = q 1 ’ 2 ( x - x#i2 + O [ ( x - xl)3’2],

6 2 i 3 M o d e r n E l e c t r o n i c S t r u c t u r e T h e o r y

d Q ( O ) - - d x

c q 1 i 2 ( q - xp2 + O [ ( X l - x)3’2] ,

Q(O) = ;q1~2(xl - x)3i2 + o[(xl - x)~/~],

dS(l) 5 _l,2

z--w z q ( x - x r ) - 5 1 2 ,

dQ(') 5 -1/2 -512 - d x

y j ! l xl- ( 4 ( 1 6 5 )

B o t h S ( l ) a n d Q ( l ) a r e i n f i n i t e a t x l .

T h e a b o v e e q u a t i o n s a r e c o n s i s t e n t w i t h

Q(x,h) = iS(xl + e i r ( x l - x),h), w3

w h i c h w o u l d i m p l y t h a t i f t o t h e l e f t o f t h e t u r n i n g p o i n t t h e w a v e f u n c t i o n

W i 3 S

t h e n t o t h e r i g h t i t s h o u l d b e

B u t E q . ( 1 6 7 ) a p p e a r s t o b e r e a l , w h e r e a s E q . ( 1 6 8 ) a p p e a r s c o m p l e x .

M o r e o v e r , t h e c o r r e s p o n d e n c e ,

Q(x, h ) = - i S ( x l + e m i r ( x l - x), h) , W%

i s a l s o c o n s i s t e n t w i t h t h e s a m e e q u a t i o n s , w h i c h w o u l d a p p e a r t o i m p l y

t h a t t o t h e r i g h t t h e w a v e f u n c t i o n s h o u l d b e n o t ( 1 6 8 ) b u t i n s t e a d

- 1 1 2

+ m e - x i / 4 e - i S ( z , h ) / h

A c l a s s i c p r o b l e m i n t h e J W K B m e t h o d i s t o c o n n e c t t h e J W K B w a v e

f u n c t i o n o n o n e s i d e o f a t u r n i n g p o i n t w i t h t h e J W K B w a v e f u n c t i o n

o n t h e o t h e r . T h e a n a l y t i c c o n t i n u a t i o n o f t h e w a v e f u n c t i o n a n d t h e

a n a l y t i c c o n t i n u a t i o n o f t h e a p p r o x i m a t e ( a s y m p t o t i c ) w a v e f u n c t i o n a r e

n o t n e c e s s a r i l y t h e s a m e . O n e s t a n d a r d m e t h o d o f s o l v i n g t h e c o n n e c t i o n -

f o r m u l a p r o b l e m i n v o l v e s u s i n g A i r y f u n c t i o n s , w h i c h a r e t h e s o l u t i o n o f

E x a c t E x p a n s i o n M e t h o d s f o r A t o m i c H y d r o g e n . . . 629

t h e S c h r o d i n g e r e q u a t i o n w i t h a l i n e a r p o t e n t i a l , e . g . , i n t h e n e i g h b o r h o o d

o f a l i n e a r t u r n i n g p o i n t . T h e A i r y f u n c t i o n s a r e t h e n r e p l a c e d b y t h e i r

a s y m p t o t i c e x p a n s i o n s o n e i t h e r s i d e , w h i c h c a n t h e n b e m a t c h e d w i t h t h e

J W K B f u n c t i o n s . A m a j o r p o i n t t o b e m a d e h e r e , h o w e v e r , i s t h a t t h e

b a r r i e r s i d e o f t h e t u r n i n g p o i n t i s a S t o k e s l i n e f o r t h e A i r y B i ( z ) , a n d , i f

t h i s i s n o t r e c o g n i z e d , c o n f u s i o n c a n a r i s e .

6 . 2 , l 3 o d S u m m a b i l i t y o f t h e A h y F u n c t i o n

A s y m p t o t i c E x p a n s i o n s

T h e A i r y f u n c t i o n s A i a n d B i ( z ) s a t i s f y

d2 G A i ( z ) = z A i ( z ) ,

d 2 p B i ( z ) = z B i ( z ) .

Ai

Bi(z) Fig. 3. The Airy functions Ai and Bi(z).


B o t h a r e e n t i r e f u n c t i o n s . F o r l a r g e p o s i t i v e z , A i i s e x p o n e n t i a l l y

s m a l l , w h i l e B i ( z ) i s e x p o n e n t i a l l y l a r g e . F o r n e g a t i v e v a l u e s , b o t h a r e

o s c i l l a t o r y . P l o t s a r e g i v e n i n F i g . 3 . T h e s o l u t i o n o f t h e c o n n e c t i o n -

f o r m u l a p r o b l e m i s g r e a t l y c l a r i f i e d b y B o r e 1 s u m m a b i l i t y o f t h e A i r y f u n c -

t i o n a s y m p t o t i c e x p a n s i o n s .

T h e A i r y f u n c t i o n a s y m p t o t i c e x p a n s i o n s i n v o i v e t h e c o e f f i c i e n t s ,

W e c o m p u t e d i r e c t l y t h e B o r e 1 s u m o f t h e s e r i e s

k=O

T h e B o r e 1 t r a n s f o r m

i s a n o r d i n a r y h y p e r g e o m e t r i c f u n c t i o n , w h i c h p r o v i d e s t h e a n a l y t i c c o n -

t i n u a t i o n t o t h e p o s i t i v e r e a l a x i s , b e y o n d t h e r a d i u s o f c o n v e r g e n c e o f t h e

h y p e r g e o m e t r i c s e r i e s ( 1 - i t i < 1 ) . T h e L a p l a c e - t r a n s f o r m s t e p t h a t g i v e s

t h e B o r e 1 s u m f r o m t h e B o r e 1 t r a n s f o r m t h e n g i v e s b y d i r e c t , n o n t r i v i a l

c o m p u t a t i o n , 3 6 l 3 7

1 ; - $ - 1 & ( 1 7 5 ) )

= ‘ _ & ? I 2 ( i < ) r ’ 6 c c A i [ ( - 3 < r ’ 3 ] . ( 1 7 6 )


Note that the hypergeometric function #r (i, $!; 1; CC) has a branch point

at z = 1. The integration path in Eq. (175) avoids the branch cut so long

as 1 arg( ~~~*‘)~ < 7r. Since t is positive, the requirement is that 1 arg [I < T.

We have thus demonstrated the following result: In the sector I arg ,z[ <

.2~/3, Ai is the Bore1 sum of the asymptotic expansion,

I arg .zl < 2~/3,

where

In the sector I arg zj < 27r/3, Ai is exponentially decreasing with z.

The expansion for the oscillatory sector follows with little effort from the

identity,

Equation (177) can be used for both of Ai(ze*r’13) provided that both

I arg .zer’j3 I < 2~/3 and I arg ze- 43 < 2~/3 are simultaneously true, i.e., 1

I arg ,zj < ~/3. One gets

k=O

2 ck(--i)k<ek,

k=O

1 axgzl < 7r/3.

(1W

The Borel-summable asymptotic expansions for the Airy Bi function simi-

larly follow from

Bi(z) = FiAi(z) + 2e*T”6Ai( ze*2ri’3) , WJ


There are three distinct domains:

00

Bi(z) = r - l / 2 z - l / 4 e + C x ck<-k 2

+ & W z - 1 / 4 e - c & k ( - < ) - k ,

k = O k = O

(0 < ax-g z < 2~/3), P w

Bi(-z) N 1 gr

- l / 2 z - l f 4 e - i ~ - ~ i / 4 I5 5

. k - k c k a

k = O

+ ~~-~12z-~l~eK+~~l~ Fck(_i)kc-k, iarg,z/ < 7r/3. L

Figure 4 sketches the domains of validity for each of the asymptotic ex-

pansions in Eqs. (177), (HO), and (183)-(185). Note in particular that the

boundaries are the Stokes lines of the expansions, at which the coefficients

Domains for Ai Domains for Bi(z)

expansions expansions

F i g . 4 . T h e d o m a i n s o n w h i c h t h e v a r i o u s a s y m p t o t i c e x p a n s i o n s f o r t h e A i r y f u n c t i o n s a r e B o r e 1 s u m m a b l e .


of the exponentially small subdominant expansions change discontinuously,

and that these lines are also cuts of the Bore1 sum formula. Note also, that

the positive axis for Bi(z) falls on such a Stokes line.

Only one formal asymptotic power series, always multiplied by an expo-

nential, occurs in the Airy function expansions. The formulae can be made

more concise by defining the basic expansion p(c) to be

k = O

C~ = I’@ + 1/6)I’@ + 5/6)/I’(l/6)I’(5/6)2’k!.

P w

Then the asymptotic expansions for Ai and Bi can be written,

Bi(z) = 7rV l’2z-1’4[p(<) - +?(-<)I, (-2~/3 < arg z < 0) ,

W J ~

Bi(z) = 7r- 1’2z-1’4[p(<) + +(-<)I, (0 < arg .z < 2~/3),

]argz] <X/3.

(191)

6.3. JWKB for a Linear Potential: JWKB Form for

Airy Function Asymptotic Expansions

The connection formulae at a linear turning point are simplest to under-

stand when the potential is linear everywhere. Then the JWKB solution

can be found term-by-term in closed form and compared with the asymp-

totic expansions given above.


Consider the Schrodinger equation with a linear potential,

The general solution is the linear combination,

$(x) = 2~1/2~-1/6~Ai[(2~q~-2)1’3(~ - xc)]

+ 2~1/2~-1/6~Bi[(2~q~-2)1’3(~ - xc)], u w

where xu = l.Z/q, and the constant factor 27r1121iM1/6 has been inserted

to simplify the final equations below. Without loss in generality, we set

2 m q = 1 and xa = 0 to make the formulae more readable.

The JWKB form of the solution involves, for instance,

(dQ/dx)m1/2eQ’h =

where (cf. Eqs. (157) and (159))

tjQ@)/& z - i x - l j 4 (QZx)2 x-l/4 = - - - x 5 -512

2 ,

Q(l) = Ax-312

48

In fact? it is not too difficult to see that


and, moreover, that Eq, (195) is essentially a rewriting of the expansion

@(c/h). That is,

--I/2

a2k-l< -2k+lfi2k

k=l

00 -112

(2k - l)a2&r<-2kh2k k=l

= (d[/dx)-1’2 /3 (C/h) . PW

This means that the coefficients &Jk-1 can be obtained recursively from

fj% - l)a2k-l<v2kh2k

k=l

ck<kii-k. PW

k=O

Given that the first several ck coefficients are explicitly

5 385 85085 Q-J = 1, Cl = yj?

-- ” - 10368’ ” = 2239488’

37182145 5391411025

” = 644972544’ ” = 46438023168 ’ (203)

one finds that the first few f& coefficients are

5 1105 82825 al = ~~ a3 = 31104’ as = 746496 *


The important result is that the JWKB expansion and the Airy asymp-

totic expansion are just rearrangements of one other: that is, they are the

same. The correspondences are

@ Q / W - 1 ’ 2 e * Q i ’ = (d[/dx)-1’2 F (*c/h) , ( 2 0 5 1

( -dS/dx)-1’2 ,Y*~‘/~ = ( -d<‘/dx)-1’2 p (Ai<‘/@ , G w

where

For the eigenfunction of

give the JWKB expansions.

Eq. (194), the expansions of Eqs. (187)-( 191)

@(x) = 27r 1’2h-1’6aAi(/i-2’3x) + 27r i/zfi-r/sbBi( h-g/sx)

N (u + ib) (dQ/dx)- 1’2 cQjfi + 2b (dQ/dx)-1’2 eQih,

(0 < argx < 2~/3)

N (u - ib) (dQ/dx)- ‘I2 cQih + 2b (dQ/dx)-1’2 eQih,

(-2~/3 < asrgx < 0)

m (b - ia) (-dS/dx)-‘i2 eisfh+ri/4

+ (b + ia) (-dS/dx)-‘i2 e-is/h-‘+,

( - 743 < arg(-x) < 43) .

Note that the expansions (209)-(211) are summable uniquely to $(x) in the

indicated sectors. Moreover, they are a statement of the connection formu-

lae to link the JWKB functions in the three sectors. The rays that bound

the sectors are Stokes lines of the expansions, across which the coefficients

of the subdominant expansions changed discontinuously. Note particularly

that the positive x axis is such a Stokes line. What is the JWKB expansion

on x > O? The answer is in part a matter of taste. From the point of view

of analysis, one must deal with Imx # 0, then take the limit Imx --+ 0.

That is, the positive x axis should be viewed by analytic continuation from

above or below. Once the choice is made, the correct expansion follows.


If the linear potential is decreasing to the right, rather than increasing,

then Eqs. (205)-(211) get replaced by

(dS/dx)- 1’2 e*isjh = (d</dx)-1’2 p&</h), @W

C -dQ/W -1’2 e*Qjh = (-d<‘/dx)-1’2 @&[‘/ti) , (213)

<’ z ;(-x)3/2 , ( 214)

$(x) =I 2~1/2~-1/6~Ai(-~-2/3x) + 2~1/2~-1/6~Bi(-~-2/3x) (215)

N (a + ib) (-dQ/dx)- lj2 e-Q/h + 2b (-dQ/dx)-‘12 eQif

(0 < arg(-x) < 2~r/3)

N (a - ib) ( -dQ/dx)-1’2 e-Q’h -I

(-2~/3 < arg(-x) < 0)

2b ( -dQ/dx)-1’2 eQih,

(-~/3 < arg x < 7r/3) . Gw

6.4. JWKB Connection Formulae via Airy Function

Asymptotic Expansions

Finally, in this section we sketch a derivation of the connection formulae

when the potential is linear only locally near the turning point. The con-

nection formulae themselves are identical with Eqs. (209)-(211) when the

barrier is to the right, and with Eqs. (216)-(218) when the barrier is to the

left.

The first step of the derivation is to use the Langer-Cherry38$3g form of

the wave function, which is designed to give a uniform asymptotic expansion

in the neighborhood of a linear classical turning point xi. For definiteness,

we take the barrier to the left of the turning point.

2?7z(JY - V) = q(x - Xl) + q(x - xl)2], 007 ( 21%

Y/J(X) = 27r1~2hM1~6 (dq5/dx)w1’2 [aAi(--hS2i3$) + bBi( --K2i34)] ,

WV


T h e f u n c t i o n s c # I ( ~ ) ( c c ) c a n b e d e t e r m i n e d r e c u r s i v e l y b y q u a d r a t u r e s

f r o m e q u a t i o n s o b t a i n e d b y s u b s t i t u t i n g E q s . ( 2 2 0 ) a n d ( 2 2 1 ) i n t o t h e

S c h r o d i n g e r e q u a t i o n . T h e i m p o r t a n t f a c t s a b o u t t h e m a r e 4 ’ t h a t t h e y

a r e a l l a n a l y t i c a t ~ 1 , a n d t h a t # ‘ ) ( z r ) = 0 ,

T h e s e c o n d s t e p o f t h e d e r i v a t i o n i s t o r e p l a c e t h e A i r y f u n c t i o n s b y

t h e i r a s y m p t o t i c e x p a n s i o n s i n p o w e r s o f I ~ c $ - ~ / ~ . F o r i n s t a n c e ,

T h e t h i r d s t e p i s t o e x p a n d e a c h p o w e r o f C $ i n p o w e r s o f i i a n d r e a r r a n g e

t h e t e r m s i n t h e s e r i e s t o h a v e J W K B f o r m .

I f i t i s p o s s i b l e t o g o f r o m E q . ( 2 2 2 ) t o ( 2 2 3 ) , t h e n

O n e f i n d s , f o r i n s t a n c e , t h a t


It is still necessary to verify that if S is defined by Eq. (224), then Eq. (223)

is valid. To this end we compute from Eq. (224)

(dS/dx)-

where we have evaluated the long numerator in Eq. (227) essentially as

the Wronskian of two Airy functions. The simplest version follows from

Eqs. (187) and (189),

p(--&-p(c) - ,8($$?(--[) = 2r(Ai(z)Bi’(z) - B@)Ai’(z)) = 2. (229)

In short, the JWKB wave function and the Langer-Cherry wave function

are rearrangements of one another. In this manner, it is clear that the

JWKB connection formulae themselves are identical with Eqs. (209)-(211)

when the barrier is to the right, and with Eqs. (216)-(218) when the barrier

is to the left.

Cne final remark is in order: The explicit relationship (224) between the

Langer-Cherry 4 and the JWKB S determines what cannot be determined

from the JWKB equations alone - particularly Eqs. (154) and (155) -

the integration constant. Equation (224), coupled with the knowledge that

$(‘) has a simple zero at the turning point, and that 4 is analytic at the

turning point, implies that S has a square-root branch point at the turning

point. In all the examples given here, that implies there is no additive

constant. Because of the exponential way that S enters the JWKB wave

function (150), an arbitrary additive constant would affect the positive

and negative exponential components differently and would make constant-

dependent modifications to the connection formulae. Fortunately, such

arbitrariness is excluded.


Acknowledgments

The author would like to thank Professor Thomas H. Bergeman for kindly

providing the experimental points and standard deviations for the exper-

iment plotted in Fig. 1, and Professors Karl H, Welge and Horst Rot tke

for kindly providing a copy of their experimental spectrum, which appears

here in Fig. 2(a).

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(1991). 23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,

U. S. Natl. Bur. Stand. Appl. Math. Ser. 55 (U. S. GPO, Washington, D.C., 1964).

24. H. J. Silverstone, J. G. Harris, J. CZek and J. PaIdus, Phys. Rev. A32, 1965 (1985).

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26. E. Borel, Ann. Sci. EC. Norm. Sup. (Paris) 16, 9 (1899).


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H. J. Silverstone in Proc. 1988 NATO Advanced Study Institute: Atoms in

Strong Fields, Kos, Greece, eds. C. A. Nicolaides, C. W. Clark and M. H. Nayfeh (Plenum, New York, 1990), p. 295.

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33. To convert from reduced-mass atomic units to laboratory units, use 1 atomic unit of energy = 2 x 109 677.583 cm-‘, and 1 atomic unit of electric field strength = 5.136 61 x 10’ V/cm.

34. H. J. Silverstone, Phys. Rev. Let& 55, 2523 (1985). 35. The discussion of the asymptotic expansions in the computation of the dis-

continuity A& given here is sharper than in Ref. 15, primarily because of the Borel-summability-based results of Ref. 34.

36. H. J. Silverstone, S. Nakai and J. G. Harris, Phys. Rev. A32, 1341 (1985). 37. M. Gailitis and H. J. Silverstone, Theovet. Chim. Acta 73, 105 (1988). 38. R. E. Langer, Phys. Rev. 51, 669 (1937). 39. T. M. Cherry, Trans. Am. Math. Sot. 68, 224 (1950). 40. In Ref. 34, the statement that follows Eq. (8) is misworded. Only rj(‘) need

vanish at the turning point.

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