I ~ ~ O C . PHYS. SOC., 1 9 6 6 , VOL. 88

~uantum defect theory [: General formulation

M. J. SEATON Department of Physics, Umversity College London

MS. received 16th March 1966

Abstract. Quantum defects provide a measure of the differences between energy levels of many-electron atoms and energy levels of hydrogenic systems. The theory for central potentials has been developed by Hartree in 1927 and 1929, Ham in 1955, and Seaton in 1955 and 1958.

The present paper is the first in a series concemed with a many-channel generalization of quantum defect theory. Wave functions Y(N+l) for an atomic system containing N+l electrons are expanded in terms of products Y&V)F, where t h e Y,(N) are functions for the corresponding N-electron system and the F, are functions for the added electron. The Fa satisfy coupled differential equations with potentials

2 being the nuclear charge. If Yu(N) or Y&V) is a bound state, U,@ goes to zero faster than l/r as r +- a. The present discussion is restricted to systems for which 2 - N > 0 and the simplifying assumption is made that Uu5 = 0, for 7 > 7 0 where r0 is &&e.

For all values of the total energy E the asymptotic forms of the functions F a can be expressed in terms of a scattenng matrk S. It is shown that S can be expressed in terms of known functions and of a matrix 11-1 where the matrices I and J are analytic functions of E. Once the matnx IJ-l is known one may calculate the bound-state energy eigenvalues, allowing for configuration interaction and series perturbations, the asymptotic forms of the normalized bound-state eigenfunctions, elastic and inelastic collision cross sections, and the positions and shapes of resonances in cross sections.

1. Introduction

dues are For an electron moving in an attractive Coulomb potential - 4 7 the energy eigen-

in atomic units, n being an integer. For a modified potential V(r), which tends to the coulomb form - z/r in the limit of 7 large, the energy eigenvalues E, may be expressed

terms of effective quantum numbers v,,

The quantum defect

1 z2 E, = - --. 0 _. 2 .& vn-

pn = n-v ,

Provides a measure of the difference between the potential V(7) and the Coulomb Potential -,z/r. It has long been known empirically that quantum defects are slowly

801

802 M. J. Seaton

varying functions of energy for simple systems and therefore provide a convenient means of summarizing information on energy levels obtained from spectral series.

An early theoretical discussion was given by Hartree (1927, 1929) and further developments in the theory have been made by Ham (1955) and by Seaton (1955, 1958 a). The work of Ham was concerned with the use of quantum defect data to obtain information about asymptotic forms of wave functions at negative energies other than the eigenvalues, and with applications to solid state problems. In the work of Seaton (1955, 1958 a) it was shown that quantum defect data could be extrapolated and used to obtain phases of wave functions for positive energies. These phases are of interest for collision problems (Seaton 1958 b), and for the calculation of photoionization cross sections (Burgess and Seaton 1960) and free-free transition probabilities (Peach 1965).

The work referred to so far has been concerned with the theory for a single electron moving in a central potential. In a more complete treatment of the many-electron problem it should be possible to account for configuration interaction and series per- turbations, resonances and autoionization, and inelastic collisions. A many-channel theory for potentials which go to zero faster than r - l in the asymptotic region has been developed by Ross and Shaw (1961) and applied to problems of e-H scattering by Damburg and Peterkop (1962). A similar theory for potentials behaving asymptotically as -z /r has been given by Gailitis (1963), who considers only positive energies, and an outline of the theory for both positive and negative energies has been given by Bely et al. (1963). In the present paper we give a more complete account of the general theory. Further papers in the present series will be concerned with applications of the theory. In the paper by Seaton (1966, to be referred to as 11) a number of illustrative applications are discussed, the paper by Bely (1966, to be referred to as 111) deals with +He+ collisions and the paper by Moores (1966, to be referred to as IV) with series perturba- tions and autoionization in calcium.

2. Coulomb functions

For an electron in a potential - z/r the Schrodinger equation in atomic units is

Putting 1 w = Y”4 +);Y(r)

-I.--- wIIc.Ic Ylm is a spherical harmonic, we obtain the rd ia i equation

a 2 Z(Z+l) 22

We put

(2.4) 1 p = zr, 2E = z2€, E = - -

K2

and replace 1 by A, where h is not necessarily an integer, to obtain

(2.5)

Quantum defect theory: I 803

webintroduce seven functions which are solutions of this equation, Y* (K , A; p), i = 1 to 7. ne fist five of these have been discussed by Seaton (1958 a) and the functions y s and y , $e required in 111. Our seven functions are

K - 1

Y2 = - Mr, - A - t t ) r(-2A)

y6 = got1 +Y2) 1

Yl = z(Y1-Y2)-

The functions and W,,,+ a are defined by Whittaker and Watson (1946),

r(.+A + 1) ' ( ~ 9 A) = K 2 1 + 2 r ( K - A )

(2.10)

(2.11)

(2.12)

(2.13)

and i a

2v ah G(K, A) = --.A(K, A)

which is absolutely and uniformly convergent. The definition of y1 may be written

(2.15)

(2.16)

where the series converges absolutely and uniformly. Since r (h+ 1 - K + O)/F(A+ 1 - K )

is a polynomial in K of order CJ, y1 may be expressed as a power series in 1 / ~ . By hmner's transfomation yl( - K, A ; p) = Y ~ ( K , A ; p) and therefore only even powers Of will occur. It follows that y1 is an analytic function of E . Furthemore, since

y2(4A;p) =Yl(K, - A - I ; p ) (2.17)

it follows that y2 is an analytic function of E.

804 M. J. Seaton

2.2. Functions for h not an integer For h # i where 1 = 0, 1,2, ... the functions y1 and y2 are hearly independent and

are real so long as h is real. I n I1 it will be necessary to consider values of h which are complex and such that the real part of h is equal to - 4. In this case the functions y6

and y, are analytic functions of E and are real and linearly independent.

2.3. Functions for h equal to an integer Letting h + 1 we obtain

so that y1 and y2 are no longer linearly independent. The function y 3 is introduced by analogy with the Weber function as a second solution of Bessel's equation of integer order. In the limit of h + 1,

From (2.13) we obtain I

A ( K , 1) = K-21-2 n (K2-p) p = o

which is a polynomial in E = - I / K ~ of order 1. The function

(2.19)

(2.20)

(2.21)

is not an analytic function of E . The analytic function y 4 ( ~ , I ; p) is obtained from 3'3

on subtracting the term involving 2Aj2h. For h = I the two linearly independent analytic functions of E are yl and 1'4. In

the remahder of &e presect ppe: we shd! consider ody &e case of r, = 1. We s h d find it convenient to use the notation

g(., I ; P ) = YdK, i ; P ) Si., i; p j = Y ~ ( K , i; p j l

2.4. Asymptotic forms We express asymptotic forms in terms of functions #* defined by

(2.22)

When E is positive we put K = iy, E = +- 1 ( E > 0) (2.24)

Y2

Quantum defect theory: I 805

,&re y is real and positive, and when E is negative we put

&rev is real and positive. We obtain +*(iy, I; p) = y'J2 exp

T U and

+*(Y, I ; p ) = v1/2exp[ - iy{k(Z-v)+$}]( ; ) 9T exp( T :). (2.27)

In the limit of p + CO, f and g cease to be single-valued functions of K. We find it

K = v + i y (Y > 0 , y > 0) (2.28)

and to take K = Y and K = ;y as the limits y - t O and v + O . Forf and g we obtain (Bateman Manuscript Project 1953, p. 278)

convenient to obtain asymptotic forms for

(2.29)

}. (2.30) (G+A cotm)C- (G+iA)++ +

r ( lS1 -K) r ( i + i + K )

The asymptotic form of ys = 8 is 8 0- m e)'e-p/K.

This function will be required only for K = Y. For this case we have

(2.31)

2.5. Properties of the function G Putting

G = 9+iX xhere $9 and X are real, it is shown by Seaton (1958 a) that

0 for K = v

for K = iy.

(2.33)

(2.34)

The function %(K, I) is not an analytic function of E = - 1/2 but may be represented an asymptotic expansion. Using (2.14) and an asymptotic expansion for +(x) (Bate-

Manuscript Project 1953, p. 47) we obtain

where the BZp are Bernoulli numbers:

(2.35)

1 30'

B - -- ... . (2.36) 1 1 8 - B - - _ 1

30' =z' 4 -

806 M. J. Seaton

using (2.33) and (2.34) we obtain, for the functions occurring in (2.30),

where

and c o t m for K = v

{-i for K = iy. C(K, I ) =

2.6. Bound states

The function f vanishes at limit of p + CO. The equation

1 d2

(2.37) (2.38)

(2.39)

(2.40)

the origin and the function O(v, I ; p ) vanishes in the

I(Z+l) 2 1 P 3

id,. -pz+-- -}y = 0 (2.41)

has solutions which are everywhere bounded only if v = n where n = I t 1 , l t 2 , ... . We take P = y(n, I ; p ) to be a bound-state solution which is real and which satisfies the normalization condition

S;P2(P)dp = 1. (2.42)

This function is P = e(%, I; p)K(n, I ) (2.43)

where

3. Description of atomic systems

We consider an atomic system consisting of a nucleus of charge 2 together uith N+1 elections. Our present interest is in bound states and in states for which one electron is in the continuum. We take the complete wave function to be

where P, is a wave function for the N-electron system, coordinates 1,2, ..., N , multi- plied by a spin function X ~ ( O " + ~ ) and a spherical harmonic Yl,,cr(BN+l, &+I), for the electron N + 1.

Substituting Y in the Schrodinger equation we obtain for the functions F, equations of the form

d2 IU(ZU+l) 22 (3.2)

where z = 2-N. In the present paper it is assumed that z > 0. The t o d energ is

E = .9gU + &&U (3.3)

z2gU being the energy for state Q. in the N-electron system.

Quantum defect theory : I 807

If GI, or s, is not a continuum state of the N-electron system it may be shown that

(3.4)

&re m Z 3 for a = /3 and m Z 2 for c( # /? (Castillejo et al. 1960). We do not con- ider the more complicated case for which a and s are both continuum states (see Rudge ad Seaton 1965). For electron collisions with neutral atoms we have z = 0 and the long-range parts of the potentials U,, may be of importance. It is shown by Gailitis ad Damburg (1963) that potentials behaving asymptotically as l/rz, connecting degener- ate states in hydrogen, produce a modification in threshold laws for electron-hydrogen mllisions. In the presence of long-range Coulomb potentials, z > 0, the asymptotic forms of the potentials Ua, are of less importance. It is shown by Gailitis (1963) that for this case no modification of threshold laws results from putting

UaB = 0 for r > ro (3.5) where ro has some large but finite value. Equation (3.5) wil l be assumed in the present paper. The effect of long-range interactions between degenerate states of hydrogenic ions will be discussed further in I11 (Bely 1966) where it will be shown that better results are obtained in practice if these interactions are taken explicitly into account. Further discussion of long-range interactions will be given in a later paper in the present series.

4. Solutions of the coupled equations

the equations (3.2) may be written (D2+W+e)F = 0.

Our convention is that quantities in italic type without subscripts represent diagonal matrices; thus E stands for the diagonal matrix with elements easa,+ For matrices which are not necessarily diagonal we use bold-face type.

Putting fa = f ( K a , la; PI, g a = g ( K a , 4; p) (4.3)

(4.4)

and assuming (3.5) we have ? - I \ A - / \ @ I n f \ n / ran\f) = Jrr\P)-"anT6a\PJV an for r^ > 90

where po = mo and where Sen, %Yan are independent of p. In (4.4), n is used to indicate the particular solution being considered. In matrix form (4.4) is

F = f$+@. (4.5) Let F(E; 9) be a solution which is zero at the origin, F(E; 0) = 0, and which is an

a y t i c functlon of E in the neighbourhood of the origin. The potential U is such that p2U + 0 as p + 0 and from this it follows that the functions Fa go to zero at least asfast p l a + l at the origin. In order to fix ideas, we could take Fan(E; p) to be such that

2

808 M. J. Seaton

or, in matrix notation,

(4.7)

( 4 4

lim (p-'-IF(E; p)) = 1. P -0

We suppose that F(E; p) is a continuous function of p and that

F(E; p) = f I +gJ for p > po.

Assuming U to satisfy certain continuity conditions, Ham (1955) shows that F(E; ,,) will be an analytic function of E for all finite values of p, and hence that IQ and J ( E ) will be analytic functions of E. The proof given by Ham is for the one-channel problem but is readily generalized.

The matrix U is Hermitian and, with the usual conventions for the choice of phases of atomic wave functions, U will be real. We may then take F(E; p) to be red and I and J to be red. Evaluating

1: [F(D2 + W + E ) F - {(D2 + W + E ) FITF]dp

where p1 > po and where F is the transpose of F, we find that

ITJ = JTI

and from t h i s it follows that IJ-1 is symmetric. Using (2.29), (2.30) and (2.37), (2.38)

(4.9)

K-')'{I-(9+iB)J). (4.10) K~ + * exp{ - i&( Z-

-++ r(z+i+K) We now introduce a second function matrix F( S ; p) such that F( S ; 0) = 0 and

F ( S ; d p : : , 4 - - 4 + S . (4.11)

Comparing (4.10) and (4.11) we see that

(4.12)

This expression, which gives the scattering matrix S in terms of the analytical matrices I and J, is the key result of the theory. The matrices I and J correspond to a function F(E; p) with some arbitrary choice of normalization, such as (4.7). The essential quantity is IJ-I. This may be represented as a power series in E together with simple poles at the zeros of the determinant IJI.

For practical calculations it is convenient to introduce Y = (B-IJ-')-'. (4.13)

Since 1J-l is real and symmetric and 59 is real and diagonal, Y is real and symnteuic* The matrix Y can be represented by asymptotic energy expansions together with sWle poles.

w Ozccmtum &fed theary: I 809

It is also convenient to introduce

R = g1/2yB1/2 (4.14)

&re B is defined by (2.39). This matrix is symmetric and can be represented by ,pptotic expansions together with simple poles. From (2.20) to (2.39) we see that B, positive so long as E , > - l/Z2 but that B, may be negative if€, < - l/Za2. It follows

that R is real for energies such that E, > - l/Za2 for all CI, but that the off-diagonal ele- ments, R,B for GI # p, may be complex if E , < - l/Z2 or eB < - l/ZB2. Also RaB = 0 for Ea = - l/Z2 or EB = - I/ZBz.

Expressing S in terms of R we obtain

S = a(l + i R ) ( l + C R ) - l b (4.15)

There the definitions of B, C, a and b may be summarized as follows:

K, = v, Kcc = '&a

B, = A, 1 - exp( - 2n y,)

c, = --2 cot rrv,

a, = eXP (4) ( 2 7 ~ / i ) ~ ' ~ exp(ih(v, - ZJv,K, (2n/i)lI2 exp{ib(v, - Z,)}v,K,

1 - exp(2rriva) b, = exP(iaa)

and where G, = argI'(Z+l-iy,) (4.16)

K, = {va2r(z,+v,+ i)qv,-z,)}-1/z. (4.17)

For most practical applications we shall use expansions containing only a few terms ; tpically we might fit to three or four terms. In some cases it is permissible to use expansions of R but this is unsatisfactory if any E , is close to - 1/Z$, or less than - 1/12. Also expansions of R cannot be used for large positive energies since the factor (l-e-2ny)-1 in (2.39) cannot be represented by expansions in powers of the energy; for small positive energies this factor is close to unity and can be neglected.

It is generally better to use expansions of Y or, in the vicinity of a pole in Y, expan- sions of Y-l. Expansions involving poles may be avoided by ihtroducing a matrix q defined by

Y = tan-. (4.18) 'his, ~II &e Q X - C ~ Z & m e , I pole i~ Y corresponds to a half-integer value of n. To calculate q in the many-channel case one diagonalizes Y,

Y = vYv-1

q = vsjv-1 where Y is diagonal, and obtains

(4.19)

(4.20) \ h e 4, = arc tan Fa. The quantity r] was introduced by Ham (1955) for the one- c ~ e l problem and referred to as the 7 defect.

In the one-channel problem the quantum defect is defined at the eigenvalues by (1.2)md(1.3). A definition ofp for aIl finite energies in the one-channel problem was given

Seaton (1958 a). Generalizing to the many-channel problem we have

tanzy = A1''YA1I2. (4.21)

810 M. J. Seaton

When all channels are closed we have B = A and tan ~p = R.

5. AU channels open

When all channels are open ( ~ = = i y = for all a),

S = eZu(l + i R ) ( l - i R ) - l efu. (5.1) Let us introduce functions

and let the function matrix F(W) have asymptotic form

F(W) N s + c W . (5.4) D 4 C

Comparison of (5.4) with (4.11) shows that the relation between the scattering matrix S and the reactance matrix W is

S = eZu(l + iB)(l - iW)-l etU (5.5) and comparison with (5.1) shows that W = R for the case in which all channels are open.

A phase matrix 6 may be defined by

R = tans. (5.6)

(5.7)

In terms of the matrix defined by (4.21) we have, for all channels open,

tannp = (1 -e-2ny)1'2 tanS(1 -e-Zny)1'2.

This is the matrix generalization of the relation between quantum defects and phases obtained by Seaton (1958 a).

6. All channels closed We put

P(p) = F ( S ; p ) S - l N (6.1) N + - S - ' N - + + N (6.2)

where N is a column vector. For closed channels 4- increases exponentially in the asymptotic region. For P to represent a bound state we must therefore have

S - ' N = 0. (6.3)

(6.4)

X = ( l + i R ) - l a - I N (6.5)

Using (4.14) with C = cot m, (6.3) may be written

Putting

(6.4) will be satisfied if

b-l co tm( tanm+ R)(1 + i R ) - l a - l N = 0.

( tanm+ R ) X = 0 (6.6)

lihich requires that jtannv-i- R I = U. (6.7)

For the one-channel case we have R = t a n r p and the solution of (6.7) is I' = tz-p uhere n is an integer, in agreement with (1.3).

From (6.5),

and using (6.6)

The bound-state functions have asymptotic form

N = ~ ( l +iR)X (6.3)

N = u(l-i tan7il ' )X. (6.9)

P - -4-N. (6.10)

It is convenient to express P in terms of the real functions 8. I3y analogy with (1.43)

P = 6KZ ( p > p o ) . (6.11) let us put

I'sing (2.32) and (6.9) we obtain

where z = 2iq- lX

1,'2

q = (-I)'(-$) COS".

For bound states R = iZ1 2YA1'2. Let us define a diagonal inatris 7 = -4-l tannv.

Using (6.6) we obtain

and hence

The condition (6.7) is equivalent to

( T + y)A1"X = 0

(T+Y)AA1'2qZ = 0.

jTfY! = 0 .

(6.12)

(6.13)

(6.14j

(6.15)

(6.16)

(6.17)

7. Normalization

We take F to be an eigenfunctian for total energy E and F' for energy E': (D2+W+e)F = 0 (7.1)

(D2+W+e')F' = 0 (7 .1J )

e@' -cc -(E' - E ) (7.2) where, by (3.3),

2 22

is independent of 51. Writing Ft far the Hermitian conjugate of F, D i

0 1 [Ft(D2+W+~')F'-((D2+W+~)F)+F']dp = 0. (7.3)

Integrating by parts, assuming F and F' both to be zera at the origin and using the Hermitian property of W, we obtain

(E-E') r1 FtF'dp = (Ft(oF')-(DF)tF'~:,,,;. (7.4) 0

812 M. J. Seaton

We now take F = F(S), F’ = F’(S). Using (2.23),

1 D+* N T+* K

and therefore 1

(€4) FtF’dp Dym (4- -+i)t7(+- +++S)’ K 0

1 -(+- +++S)t-(+- -++S)’.

K t

(7.5’

(7.6)

Consider first the case in which all channels are open. Putting e’ = -E in (7.6) obtain

For E‘ # E we consider J,” FtF’dp as a factor occurring in the integrand of an integral over €4. After some reduction we obtain

StS = 1. (7.7)

[IFtF’dp = ~&(E--E’).

If some channels are open and some closed, (7.8) remains valid for matrii elements connecting open channels.

When all channels are closed we put

P = F S - I N with S-IN = 0 (7.9) SO that P is an eigenfunction. We put

P’ = (FS-IN)’ where, in accordance with (6.9),

N’ = a’(1-itanm’)X’.

(7.10)

(7.11)

Apart from a factor of normalization, the vectors N and X are defined at the eigenvalues. The functions P’ are solutions of (7.1’) for energies which are not eigenvalues and the vectors N’ and X‘ are therefore not yet defined. As a matter of convenience we take X’ = x.

From (7.6) we obtain 1

PtP‘dp N (++ N)t--(+-S-’++’)‘N‘ V

and, since ++ + O for p +- CO, 11 Pt Pdp = lim lim Nt(+ +)*( 1 /v’ + 1 /v)(+-)’(S N)’ (7.13) D 4 0 &’ +& E’-€

Since (S-IN)‘ +- 0 for E’-€ + 0, in (7.13) we may replace V‘ by v and (4-)’ by #-. (7.14) (++)*+- = vexp{(in(Z-v)}

and hence

BY (2.27))

2 exp{in(l-v)) PtPdp = lim Nt (S-I N)’ s: E’ +E E --E

(7.15)

Quantum defect theory: I 813

uiig (4.14) and (7.11) with X‘ = X, we have

(S-lN)’ = {b-lcotm(tanm+R))’X (7.16)

ad siice (tan m+ R)X =:O we obtain, to first order in E’-€,

(S-lN)‘ = (b-l Cotm)(E‘-E)

Snce E = - l/.” we have a

-tan 7 v = q-2 ae

(7.17)

(7.18)

where q is defined by (6.13), and hence

a aE -(tanm+ R) = q-lt;q-l (7.19)

where a R r = ‘+P(-&.

For the normalization integral (7.15) we obtain, using (7.17) and (7.20),

(7.20)

(7.21)

U h g (6.9) and (6.12) to express Nt and X in terms of Z we find, after some straight- forward reductions, that the function P = OK2 will be normalized to

s,” PtPdp = 1

provided that K* K

zt+z = 1.

(7.22)

(7.23)

- Ihe functions p may always be taken 10 be red. For v c 1, K ”ay be h-eghary but KZ will be red. Introducing the transpose of Z, ZT = (Zt)*, we have ZT = Z+K*/K for KZ real. The normalization condition (7.24) may therefore be written

Z T Z = 1. (7.24) It is convenient to express < in terms of Y. Using R = A”2YrZ”’ we ~ b t k i

and using (6.16) it is seen that, in (7.24), we may take

(7.26)

For highly excited states v will be large and q, defined by (7.19), will be small. In case we may take 6 = 1 and the normalization condition becomes zTz = 1.

Whenever this approximation is valid, Z will be real.

814 M. J. Seaton

It is sometimes found that, for the lowest energy level of a system, one has a COm- ponent for which vu -N 1,. It is readily shown that

(7.27)

and from this it follows that tU8 = 0 for vu = 1,. In the limit of vu +I,, k, and 5, behave as v,-Z,, K, behaves as ( v , - & ) ~ ’ ~ and 2, behaves as ( ~ , - l ~ ) - ~ ’ ~ . The produa K,Z, occurring in (6.11) therefore remains real and finite.

8. Discussion We have given an exact theory of the analytical properties of the solutions of equt-

ion (3.2)) it being assumed that U = 0 for r > ro where ro is finite. Although an ema theory of the many-electron problem would be more complicated, in consequence of the long range of the potential matrix U, it is believed that the present theory is adequate for most practical purposes of interpolation and extrapolation.

In I1 (Seaton 1966) we shall give some illustrative applications of the theory for the interpretation of perturbed series, resonances and collision cross sections.

Acknowledgments

about this work. I should like to thank Dr. 0. Bely and Dr. D. Moores for many valuable discussions

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(London